A385498 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n,k).
1, 6, 48, 408, 3564, 31626, 283548, 2560872, 23255964, 212101176, 1941110628, 17815257048, 163896843300, 1510891524252, 13952756564424, 129048895061208, 1195191116753436, 11082661017288264, 102877353868090080, 955912961224763232, 8889969049985302464
Offset: 0
Programs
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Mathematica
Table[(81/8)^n - Binomial[4*n, n]*(-1 + Hypergeometric2F1[1, -3*n, 1 + n, -1/2]), {n,0,25}] (* Vaclav Kotesovec, Jul 30 2025 *) -
PARI
a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n, k));
Formula
a(n) = [x^n] 1/((1-3*x) * (1-x)^(3*n)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(3*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 24*n*(3*n - 2)*(3*n - 1)*(139*n^3 - 366*n^2 + 143*n + 132)*a(n) = (588665*n^6 - 2281011*n^5 + 2262209*n^4 + 1245939*n^3 - 3359986*n^2 + 1877400*n - 322560)*a(n-1) - 648*(2*n - 3)*(4*n - 7)*(4*n - 5)*(139*n^3 + 51*n^2 - 172*n + 48)*a(n-2).
a(n) ~ 2^(8*n + 1/2) / (sqrt(Pi*n) * 3^(3*n - 1/2)). (End)
G.f.: g/((3-2*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^3*(12-6*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025