Difference between revisions of "OEIS/Coordination sequences for lattices"

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(Created page with "The recurrences were derived by Mathematica: bin[n_,k_]:=(n!/(n-k)!/k!); === C_n === * [https://oeis.org/A103884 A103884 (C_4)] d:= 4; CoefficientList[Series[(Sum[Binomial[...")
 
 
(One intermediate revision by the same user not shown)
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The recurrences were derived by Mathematica:
+
The recurrences were derived by Mathematica. Auxilliary function:
 
  bin[n_,k_]:=(n!/(n-k)!/k!);
 
  bin[n_,k_]:=(n!/(n-k)!/k!);
 +
 +
=== A_n ===
 +
* [https://oeis.org/A103881 A103881 (C_4)]
 +
d:= 4; CoefficientList[Series[Sum[(Binomial[d,k])^2*x^k, {k, 0, d}]/(1-x)^d, {x,0,11}],x]
 +
{1, 20, 110, 340, 780, 1500, 2570, 4060, 6040, 8580, 11750, 15620}
 +
InputForm[FullSimplify[bin[n,k]^2]] -> (1 - k + n)^2/k^2
 +
n:=4; f:= gfun:-rectoproc({a(k)=(n-(k-1))^2/(k^2)*a(k-1),a(0)=1},a(k),remember): map(f, [$0..n]);
 +
[1, 16, 36, 16, 1]
 +
=== A*_n ===
 +
* [https://oeis.org/A204621 A204621] coordinator triangle, rows are the g.f.s numerator coefficients
 +
* [https://oeis.org/A008535 A008535 (A*_7)]
 +
d:=7; CoefficientList[Series[((Sum[Sum[Binomial[d+1, j], {j,0,Min[k,d-k]}]*x^k, {k,0,d}])/(1-x)^d),{x,0,10}],x]
 +
{1, 16, 128, 688, 2746, 8752, 23536, 55568, 118498, 232976, 428752}
 +
=== B_n ===
 +
A103883
 +
d:= 4; CoefficientList[Series[(Sum[(Binomial[2n+1,2k] - 2*k*Binomial[d,k])*x^k, {k,0,d}])/(1-x)^d, {x,0,11}],x]
 +
{1, 32, 224, 768, 1856, 3680, 6432, 10304, 15488, 22176, 30560, 40832}
  
 
=== C_n ===
 
=== C_n ===
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  {1, 32, 192, 608, 1408, 2720, 4672, 7392, 11008, 15648, 21440, 28512}
 
  {1, 32, 192, 608, 1408, 2720, 4672, 7392, 11008, 15648, 21440, 28512}
 
  InputForm[FullSimplify[bin[2*n,2*k]/bin[2*n,2*k-2]]] -> ((-1 + 2*k - 2*n)*(-1 + k - n))/(k*(-1 + 2*k))
 
  InputForm[FullSimplify[bin[2*n,2*k]/bin[2*n,2*k-2]]] -> ((-1 + 2*k - 2*n)*(-1 + k - n))/(k*(-1 + 2*k))
  n:=4; f:= gfun:-rectoproc({a(k)=(2*n-2*k+1)*(n-(k-1))/(k*(2*k-1))*a(k-1),a(0)=1},a(k),remember): map(f, [$0
+
  n:=4; f:= gfun:-rectoproc({a(k)=(2*n-2*k+1)*(n-(k-1))/(k*(2*k-1))*a(k-1),a(0)=1},a(k),remember): map(f, [$0..n]);
  Recurrence: a(k)=(2*n-2*k+1)*(n-(k-1))/(k*(2*k-1))*a(k-1), a(0)=1, k=0..n
+
[1, 28, 70, 28, 1]
 +
=== D_n ===
 +
A103903
 +
d:= 4; CoefficientList[Series[(Sum[(Binomial[2n,2k]-2*d*Binomial[d-2,k-1])*x^k, {k, 0, d}])/(1-x)^d, {x,0,11}],x]
 +
{1, 24, 144, 456, 1056, 2040, 3504, 5544, 8256, 11736, 16080, 21384}
 +
=== D*_n ===
 +
A035706
 +
d:= 12; CoefficientList[Series[(Sum[Binomial[d,k]*x^k, {k, 0, d}] + 2^d*x^(d/2))/(1 - x)^d, {x,0,11}],x]
 +
{1, 24, 288, 2312, 14016, 68664, 288096, 1071912, 3600768, 11036504, 31125408, 81412680}
 +
===Cubic lattice===
 +
A035706
 +
d:= 11; CoefficientList[Series[Sum[Binomial[d,k]*x^k, {k, 0, d}]/(1 - x)^d, {x,0,11}],x]
 +
{1, 22, 242, 1782, 9922, 44726, 170610, 568150, 1690370, 4573910, 11414898, 26572086}

Latest revision as of 16:45, 28 July 2020

The recurrences were derived by Mathematica. Auxilliary function:

bin[n_,k_]:=(n!/(n-k)!/k!);

A_n

d:= 4; CoefficientList[Series[Sum[(Binomial[d,k])^2*x^k, {k, 0, d}]/(1-x)^d, {x,0,11}],x]
{1, 20, 110, 340, 780, 1500, 2570, 4060, 6040, 8580, 11750, 15620}
InputForm[FullSimplify[bin[n,k]^2]] -> (1 - k + n)^2/k^2
n:=4; f:= gfun:-rectoproc({a(k)=(n-(k-1))^2/(k^2)*a(k-1),a(0)=1},a(k),remember): map(f, [$0..n]);
[1, 16, 36, 16, 1]

A*_n

d:=7; CoefficientList[Series[((Sum[Sum[Binomial[d+1, j], {j,0,Min[k,d-k]}]*x^k, {k,0,d}])/(1-x)^d),{x,0,10}],x]
{1, 16, 128, 688, 2746, 8752, 23536, 55568, 118498, 232976, 428752}

B_n

A103883
d:= 4; CoefficientList[Series[(Sum[(Binomial[2n+1,2k] - 2*k*Binomial[d,k])*x^k, {k,0,d}])/(1-x)^d, {x,0,11}],x]
{1, 32, 224, 768, 1856, 3680, 6432, 10304, 15488, 22176, 30560, 40832}

C_n

d:= 4; CoefficientList[Series[(Sum[Binomial[2n,2k]*x^k, {k, 0, d}])/(1-x)^d, {x,0,11}],x]
{1, 32, 192, 608, 1408, 2720, 4672, 7392, 11008, 15648, 21440, 28512}
InputForm[FullSimplify[bin[2*n,2*k]/bin[2*n,2*k-2]]] -> ((-1 + 2*k - 2*n)*(-1 + k - n))/(k*(-1 + 2*k))
n:=4; f:= gfun:-rectoproc({a(k)=(2*n-2*k+1)*(n-(k-1))/(k*(2*k-1))*a(k-1),a(0)=1},a(k),remember): map(f, [$0..n]);
[1, 28, 70, 28, 1]

D_n

A103903
d:= 4; CoefficientList[Series[(Sum[(Binomial[2n,2k]-2*d*Binomial[d-2,k-1])*x^k, {k, 0, d}])/(1-x)^d, {x,0,11}],x]
{1, 24, 144, 456, 1056, 2040, 3504, 5544, 8256, 11736, 16080, 21384}

D*_n

A035706
d:= 12; CoefficientList[Series[(Sum[Binomial[d,k]*x^k, {k, 0, d}] + 2^d*x^(d/2))/(1 - x)^d, {x,0,11}],x]
{1, 24, 288, 2312, 14016, 68664, 288096, 1071912, 3600768, 11036504, 31125408, 81412680}

Cubic lattice

A035706
d:= 11; CoefficientList[Series[Sum[Binomial[d,k]*x^k, {k, 0, d}]/(1 - x)^d, {x,0,11}],x]
{1, 22, 242, 1782, 9922, 44726, 170610, 568150, 1690370, 4573910, 11414898, 26572086}