A371431 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2 + x^3) ).

1, 2, 7, 29, 131, 623, 3064, 15423, 78936, 408958, 2137993, 11252163, 59508232, 315786764, 1679410076, 8941421014, 47613443433, 253359512287, 1346009853489, 7133000408765, 37669665812955, 198034693198875, 1035095172883710, 5371011415598595, 27615259784888724

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1289

Index entries for reversions of series

Formula

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

D-finite with recurrence: (-14592*n^3 - 51072*n^2 - 57456*n - 20520)*a(n) + (-60643*n^3 - 309615*n^2 - 524714*n - 295560)*a(n + 1) + (10444*n^3 + 42411*n^2 - 2287*n - 106572)*a(n + 2) + (1853*n^3 + 23874*n^2 + 98557*n + 130836)*a(n + 3) + (-310*n^3 - 3720*n^2 - 14570*n - 18600)*a(n + 4) = 0. - Robert Israel, Mar 11 2026

Maple

f:= gfun:-rectoproc({(-14592*n^3 - 51072*n^2 - 57456*n - 20520)*a(n) + (-60643*n^3 - 309615*n^2 - 524714*n - 295560)*a(n + 1) + (10444*n^3 + 42411*n^2 - 2287*n - 106572)*a(n + 2) + (1853*n^3 + 23874*n^2 + 98557*n + 130836)*a(n + 3) + (-310*n^3 - 3720*n^2 - 14570*n - 18600)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 7, a(3) = 29}, 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)^2+x^3))/x)
(PARI) a(n) = sum(k=0, n\3, (-1)^k*binomial(n+k, k)*binomial(3*n-k+1, n-3*k))/(n+1);

Crossrefs

Cf. A369214.

Sequence in context: A193040 A200755 A132262 * A007852 A300048 A232971

Adjacent sequences: A371428 A371429 A371430 * A371432 A371433 A371434

Keyword

nonn

Author

Seiichi Manyama, Mar 23 2024

Status

approved