A365855 Expansion of (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^4 ).

1, 2, 9, 46, 264, 1612, 10291, 67830, 458109, 3153744, 22049065, 156127140, 1117369884, 8069610992, 58735003740, 430416574918, 3172987081311, 23514565653058, 175083678670264, 1309132916709168, 9825882638364144, 74003924059921940, 559112987425763365

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Index entries for reversions of series

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(2*n+k+1,k) * binomial(5*n-k+3,n-k).

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(3*n-2*k+1,n-2*k). - Seiichi Manyama, Jan 18 2024

a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^2 * (1-x)^4 )^(n+1). - Seiichi Manyama, Feb 16 2024

D-finite with recurrence 41287680*n*(4*n+3)*(2*n+1) *(2*n-1) *(101600834504882*n -79530504768063) *(4*n+1)*(n+1)*a(n) -2048*n*(2*n-1) *(434506669234595351728*n^5 +44263551729813272744*n^4 -141669451985575941466*n^3 -268951240644423601763*n^2 +134606020676490900438*n -20668707583222185981)*a(n-1) +8*(-398564930699101536727520*n^7 +543141971021842664782320*n^6 +837332804222429331454882*n^5 -2219966207175180712139775*n^4 +2126251388322307117458940*n^3 -1338986566696277643223185*n^2 +555014916985484581653258*n -99934175131659030433320)*a(n-2) +7*(433107331515621433*n -223378841061948614)*(7*n-11) *(7*n-15)*(7*n-12) *(7*n-9)*(7*n-13) *(7*n-10)*a(n-3)=0. - R. J. Mathar, Dec 02 2025

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k+1, k)*binomial(5*n-k+3, n-k))/(n+1);
(SageMath)
def A365855(n):
    h = binomial(5*n + 3, n) * hypergeometric([-n, 2*n + 2], [-5 * n - 3], -1) / (n + 1)
    return simplify(h)
print([A365855(n) for n in range(23)])  # Peter Luschny, Sep 20 2023

Crossrefs

Cf. A365854, A365856.

Cf. A365752, A368967.

Sequence in context: A114194 A218045 A161798 * A373312 A134091 A380497

Adjacent sequences: A365852 A365853 A365854 * A365856 A365857 A365858

Keyword

nonn

Author

Seiichi Manyama, Sep 20 2023

Status

approved