A386701 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(4*n,k).
1, 1, 13, 103, 869, 7476, 65323, 577242, 5144949, 46167196, 416527828, 3774785983, 34336862435, 313330665532, 2866982877954, 26294890918308, 241665561294741, 2225104901535564, 20520648006149980, 189523353219338572, 1752680220372189364, 16227703263403842768
Offset: 0
Programs
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Mathematica
Table[(-16/27)^n - Binomial[4*n, n]*(-1 + Hypergeometric2F1[1, -3*n, 1 + n, 1/3]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *) -
PARI
a(n) = sum(k=0, n, (-3)^(n-k)*binomial(4*n, k));
Formula
a(n) = [x^n] 1/((1+2*x) * (1-x)^(3*n)).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 81*n*(3*n - 2)*(3*n - 1)*(612*n^3 - 2838*n^2 + 4354*n - 2209)*a(n) = 24*(165240*n^6 - 1019628*n^5 + 2493432*n^4 - 3068178*n^3 + 1984652*n^2 - 632900*n + 76545)*a(n-1) + 128*(2*n - 3)*(4*n - 7)*(4*n - 5)*(612*n^3 - 1002*n^2 + 514*n - 81)*a(n-2).
a(n) ~ 2^(8*n - 1/2) / (sqrt(Pi*n) * 3^(3*n + 1/2)). (End)
G.f.: g/((-2+3*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} (-2)^k * 3^(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*(-8+9*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025