A264717 Central terms of triangle A100326.

1, 4, 46, 626, 9094, 136792, 2102728, 32804760, 517325270, 8225083124, 131614959262, 2116988791018, 34196629924584, 554369366584256, 9014333613083632, 146961155561594176, 2401364353568376054, 39316907672544234028, 644861670750937767370

Offset: 0

Author

Reinhard Zumkeller, Nov 21 2015

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A100326.

Programs

Haskell

a264717 n = a100326 (2 * n) n

Magma

[n le 2 select 4^(n-1) else ( 6*(1797120*n^8 -28080000*n^7 [phone]*n^6 -715605188*n^5 [phone]*n^4 -2421570243*n^3 [phone]*n^2 -1066134220*n [phone])*Self(n-1) +3*(3*n-8)*(3*n-10)*(2*n-5)*(n-3)*(1248*n^4 -5772*n^3 +8387*n^2 -3031*n -1158)*Self(n-2))/(16*(n-2)*(2*n-3)*(4*n-7)*(4*n-5)*(1248*n^4 -10764*n^3 +33191*n^2 -42113*n +17280)): n in [1..41]]; // G. C. Greubel, Jan 30 2023

Mathematica

a[n_]:= a[n]= If[n<2, 4^n, (6*(1797120*n^8 -13703040*n^7 [phone]*n^6 -70197188*n^5 [phone]*n^4 -29185735*n^3 +4100685*n^2 +1396683*n - 409602)*a[n-1] +3*(3*n-5)*(3*n-7)*(2*n-3)*(n-2)*(1248*n^4 -780*n^3 -1441*n^2 +1419*n -326)*a[n-2])/(16*(n-1)*(2*n-1)*(4*n-3)*(4*n-1)*(1248*n^4 -5772*n^3 +8387*n^2 -3031*n -1158))];
Table[a[n], {n,0,40}] (* G. C. Greubel, Jan 30 2023 *)

SageMath

def p(n): return 1797120*n^8 -13703040*n^7 [phone]*n^6 -70197188*n^5 [phone]*n^4 -29185735*n^3 +4100685*n^2 +1396683*n - 409602
def q(n): return (3*n-5)*(3*n-7)*(2*n-3)*(n-2)*(1248*n^4 -780*n^3 -1441*n^2 +1419*n -326)
@CachedFunction
def a(n): # a = A264717
    if(n<2): return 4^n
    else: return (6*p(n)*a(n-1) +3*q(n)*a(n-2))/(16*(n-1)*(2*n-1)*(4*n-3)*(4*n-1)*(1248*n^4 -5772*n^3 +8387*n^2 -3031*n -1158))
[a(n) for n in range(41)] # G. C. Greubel, Jan 30 2023

Formula

a(n) = A100326(2*n,n).

a(n) = (6*(1797120*n^8 -13703040*n^7 [phone]*n^6 -70197188*n^5 [phone]*n^4 -29185735*n^3 +4100685*n^2 +1396683*n - 409602)*a(n-1) +3*(3*n-5)*(3*n-7)*(2*n-3)*(n-2)*(1248*n^4 -780*n^3 -1441*n^2 +1419*n -326)*a(n-2))/(16*(n-1)*(2*n-1)*(4*n-3)*(4*n-1)*(1248*n^4 -5772*n^3 +8387*n^2 -3031*n -1158)). - G. C. Greubel, Jan 30 2023

a(n) ~ 3^(3*n/2 - 1) * (1 + sqrt(3))^(6*n + 1/2) / (sqrt(Pi*n) * 2^(7*n + 1/2)). - Vaclav Kotesovec, Jan 31 2023