A349582 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^3 * A(x)^5.

1, 3, 13, 85, 733, 7292, 78267, 880250, 10226237, 121713373, 1476272394, 18180126906, 226704989103, 2856790765238, 36321840773980, 465362291912648, 6002272018481901, 77873186277771107, 1015583616140910999, 13306207249869273003, 175064043975233981626

Offset

0,2

Comments

Second binomial transform of A002294.

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*k,k) * 2^(n-k) / (4*k+1).

a(n) = 2^n*F([1/5, 2/5, 3/5, 4/5, -n], [1/2, 3/4, 1, 5/4], -5^5/2^9), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021

a(n) ~ 3637^(n + 3/2) / (78125 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Nov 26 2021

Mathematica

nmax = 20; A[_] = 0; Do[A[x_] = 1/(1 - 2 x) + x (1 - 2 x)^3 A[x]^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[5 k, k] 2^(n - k)/(4 k + 1), {k, 0, n}], {n, 0, 20}]

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(5*k, k)*2^(n-k)/(4*k+1)); \\ Michel Marcus, Nov 23 2021

Crossrefs

Cf. A002294, A064613, A346647, A346762, A349581, A349584, A349590, A349591.

Sequence in context: A369550 A125500 A121679 * A246387 A023037 A354436

Adjacent sequences: A349579 A349580 A349581 * A349583 A349584 A349585

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 22 2021

Status

approved