A371427 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 - x^4) ).

1, 2, 5, 14, 41, 122, 363, 1066, 3046, 8300, 20791, 43738, 51297, -174406, -1825027, -10480330, -50143510, -218385772, -895007802, -3504952380, -13214355159, -48116028934, -169216483595, -573113441834, -1856620607526, -5675964306988, -15927363432481

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1695

Index entries for reversions of series

Formula

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

D-finite with recurrence: (-2160*n^3 - 12960*n^2 - 23760*n - 12960)*a(n) + (3408*n^3 + 27960*n^2 + 75048*n + 65520)*a(n + 1) + (-5828*n^3 - 56372*n^2 - 182492*n - 197484)*a(n + 2) + (4216*n^3 + 50628*n^2 + 203296*n + 273234)*a(n + 3) + (-1607*n^3 - 23944*n^2 - 119168*n - 198276)*a(n + 4) + (324*n^3 + 5724*n^2 + 33672*n + 65970)*a(n + 5) + (-27*n^3 - 540*n^2 - 3588*n - 7920)*a(n + 6) = 0. - Robert Israel, Mar 11 2026

Maple

f:= gfun:-rectoproc({(-2160*n^3 - 12960*n^2 - 23760*n - 12960)*a(n) + (3408*n^3 + 27960*n^2 + 75048*n + 65520)*a(n + 1) + (-5828*n^3 - 56372*n^2 - 182492*n - 197484)*a(n + 2) + (4216*n^3 + 50628*n^2 + 203296*n + 273234)*a(n + 3) + (-1607*n^3 - 23944*n^2 - 119168*n - 198276)*a(n + 4) + (324*n^3 + 5724*n^2 + 33672*n + 65970)*a(n + 5) + (-27*n^3 - 540*n^2 - 3588*n - 7920)*a(n + 6), a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 14, a(4) = 41, a(5) = 122}, 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^4))/x)
(PARI) a(n) = sum(k=0, n\4, (-1)^k*binomial(n+1, k)*binomial(2*n-2*k+2, n-4*k))/(n+1);

Crossrefs

Cf. A001006, A371426.

Cf. A369158.

Sequence in context: A307466 A382570 A116849 * A123183 A007051 A124302

Adjacent sequences: A371424 A371425 A371426 * A371428 A371429 A371430

Keyword

sign

Author

Seiichi Manyama, Mar 23 2024

Status

approved