A371433 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3 + x) ).

1, 2, 5, 11, 11, -77, -704, -3795, -15686, -48598, -74009, 376623, 4438840, 27458060, 126898948, 440550682, 849522927, -2621906045, -39993434701, -270428078305, -1339005344985, -5014789377825, -11407684195950, 18849058485855, 417417757017612, 3058172078113944

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1415

Index entries for reversions of series

Formula

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

D-finite with recurrence: (-768*n^3 - 2688*n^2 - 3024*n - 1080)*a(n) + (3073*n^3 + 17925*n^2 + 35318*n + 23400)*a(n + 1) + (-3374*n^3 - 28761*n^2 - 81889*n - 77916)*a(n + 2) + (925*n^3 + 9540*n^2 + 32555*n + 36780)*a(n + 3) + (-92*n^3 - 1104*n^2 - 4324*n - 5520)*a(n + 4) = 0. - Robert Israel, Mar 11 2026

Maple

f:= gfun:-rectoproc({(-768*n^3 - 2688*n^2 - 3024*n - 1080)*a(n) + (3073*n^3 + 17925*n^2 + 35318*n + 23400)*a(n + 1) + (-3374*n^3 - 28761*n^2 - 81889*n - 77916)*a(n + 2) + (925*n^3 + 9540*n^2 + 32555*n + 36780)*a(n + 3) + (-92*n^3 - 1104*n^2 - 4324*n - 5520)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 11}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 11 2026

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3+x))/x)
(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+k, k)*binomial(4*n+2*k+2, n-k))/(n+1);

Crossrefs

Cf. A371434, A371435.

Cf. A369215.

Sequence in context: A091114 A155767 A079782 * A093554 A168453 A136990

Adjacent sequences: A371430 A371431 A371432 * A371434 A371435 A371436

Keyword

sign

Author

Seiichi Manyama, Mar 23 2024

Status

approved