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

0,2

Links

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

Formula

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

a(n) = (6*(1797120*n^8 -13703040*n^7 +42834240*n^6 -70197188*n^5 +63370677*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

Mathematica

a[n_]:= a[n]= If[n<2, 4^n, (6*(1797120*n^8 -13703040*n^7 +42834240*n^6 -70197188*n^5 +63370677*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 *)

Prog

(Haskell)
a264717 n = a100326 (2 * n) n
(Magma) [n le 2 select 4^(n-1) else ( 6*(1797120*n^8 -28080000*n^7 +189074880*n^6 -715605188*n^5 +1662275017*n^4 -2421570243*n^3 +2154450632*n^2 -1066134220*n +223382400)*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
(SageMath)
def p(n): return 1797120*n^8 -13703040*n^7 +42834240*n^6 -70197188*n^5 +63370677*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

Crossrefs

Cf. A100326.

Sequence in context: A235132 A236956 A113264 * A318109 A234527 A126739

Adjacent sequences: A264714 A264715 A264716 * A264718 A264719 A264720

Keyword

nonn

Author

Reinhard Zumkeller, Nov 21 2015

Status

approved