A371428 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3 - x^2) ).

1, 3, 11, 46, 209, 1003, 5002, 25665, 134605, 718371, 3888633, 21298962, 117823660, 657344600, 3694378463, 20896495211, 118865999117, 679545095167, 3902327585407, 22499738052954, 130200110475407, 755927955655813, 4402088019958400, 25706104810367515

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1259

Index entries for reversions of series

Formula

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

D-finite with recurrence: (69*n^2 + 207*n + 138)*a(n) + (-98*n^2 - 372*n - 352)*a(n + 1) + (-462 - 144*n)*a(n + 2) + (52*n^2 + 486*n + 1130)*a(n + 3) + (-8*n^2 - 84*n - 220)*a(n + 4) = 0. - Robert Israel, Mar 11 2026

Maple

f:= gfun:-rectoproc({(69*n^2 + 207*n + 138)*a(n) + (-98*n^2 - 372*n - 352)*a(n + 1) + (-462 - 144*n)*a(n + 2) + (52*n^2 + 486*n + 1130)*a(n + 3) + (-8*n^2 - 84*n - 220)*a(n + 4), a(0) = 1, a(1) = 3, a(2) = 11, a(3) = 46}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 11 2026

Mathematica

Table[1/(n+1) Sum[(-1)^k Binomial[n+1, k]Binomial[3n-3k+3, n-2k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Harvey P. Dale, Sep 25 2024 *)

Prog

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

Crossrefs

Cf. A107264, A127897, A371429.

Sequence in context: A306822 A193074 A287891 * A233389 A281548 A086521

Adjacent sequences: A371425 A371426 A371427 * A371429 A371430 A371431

Keyword

nonn

Author

Seiichi Manyama, Mar 23 2024

Status

approved