A386702 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n,k).
1, 2, 24, 248, 2676, 29562, 331956, 3771896, 43242660, 499215146, 5795429764, 67587697872, 791232339756, 9292673328174, 109440405341088, 1291977861163968, 15284200451058724, 181147979395807002, 2150493166839159936, 25567085678133719880, 304368033788893315896
Offset: 0
Programs
-
Mathematica
Table[(-32/81)^n - Binomial[5*n, n]*(-1 + Hypergeometric2F1[1, -4*n, 1 + n, 1/3]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *) -
PARI
a(n) = sum(k=0, n, (-3)^(n-k)*binomial(5*n, k));
Formula
a(n) = [x^n] 1/((1+2*x) * (1-x)^(4*n)).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 648*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(37331*n^4 - 227972*n^3 + 518701*n^2 - 521044*n + 194928)*a(n) = (9143593823*n^8 - 74277961298*n^7 + 253684378280*n^6 - 473415527402*n^5 + 524935655069*n^4 - 351762123620*n^3 + 137998180332*n^2 - 28677229776*n + 2380855680)*a(n-1) + 160*(5*n - 9)*(5*n - 8)*(5*n - 7)*(5*n - 6)*(37331*n^4 - 78648*n^3 + 58771*n^2 - 18234*n + 1944)*a(n-2).
a(n) ~ 5^(5*n + 1/2) / (7 * sqrt(Pi*n) * 2^(8*n - 1/2)). (End)
G.f.: g/((-2+3*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^4*(-10+12*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025