A369441 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x^2)^2) ).

1, 2, 7, 30, 141, 704, 3666, 19686, 108222, 606062, 3445308, 19829680, 115323955, 676659960, 4000719012, 23811922678, 142557391306, 857894530348, 5186614665121, 31487226410770, 191871141682557, 1173163962971056, 7195329233469552, 44255915928488880

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1224

Index entries for reversions of series

Formula

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

From Seiichi Manyama, Feb 23 2026: (Start)

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (-1)^k * binomial(2*n+2,k) * binomial(3*n-4*k+1,2*n+1).

G.f.: B(x)^2, where B(x) is the g.f. of A186241. (End)

D-finite with recurrence: 59055800320*(2*n + 7)*(2*n + 5)*(2*n + 3)*(n + 2)*(n + 1)*a(n) - 251658240*(2*n + 5)*(n + 2)*(2*n + 7)*(122*n^2 + 1301*n + 2926)*a(n + 1) + 1572864*(2*n + 7)*(164592*n^4 + 1526928*n^3 + 4871252*n^2 + 5655443*n + 1013235)*a(n + 2) - 8192*(38358248*n^5 + 701271300*n^4 + 5150704730*n^3 + 18996983445*n^2 + 35172579122*n + 26139802620)*a(n + 3) - 2560*(744976*n^5 + 80125608*n^4 + 1346800360*n^3 + 9126563382*n^2 + 27992968168*n + 32399724291)*a(n + 4) + 96*(51911982*n^5 + 1364982945*n^4 + 14225596535*n^3 + 73327104330*n^2 + 186511232743*n + 186671736465)*a(n + 5) + 28*(4259633*n^5 + 139922175*n^4 + 1832397545*n^3 + 11954401995*n^2 + 38835414092*n + 50233052820)*a(n + 6) + 1015*(5*n + 39)*(n + 8)*(5*n + 41)*(5*n + 37)*(5*n + 38)*a(n + 7) = 0. - Robert Israel, Mar 20 2026

Maple

f:= gfun:-rectoproc({59055800320*(2*n + 7)*(2*n + 5)*(2*n + 3)*(n + 2)*(n + 1)*a(n) - 251658240*(2*n + 5)*(n + 2)*(2*n + 7)*(122*n^2 + 1301*n + 2926)*a(n + 1) + 1572864*(2*n + 7)*(164592*n^4 + 1526928*n^3 + 4871252*n^2 + 5655443*n + 1013235)*a(n + 2) - 8192*(38358248*n^5 + 701271300*n^4 + 5150704730*n^3 + 18996983445*n^2 + 35172579122*n + 26139802620)*a(n + 3) - 2560*(744976*n^5 + 80125608*n^4 + 1346800360*n^3 + 9126563382*n^2 + 27992968168*n + 32399724291)*a(n + 4) + 96*(51911982*n^5 + 1364982945*n^4 + 14225596535*n^3 + 73327104330*n^2 + 186511232743*n + 186671736465)*a(n + 5) + 28*(4259633*n^5 + 139922175*n^4 + 1832397545*n^3 + 11954401995*n^2 + 38835414092*n + 50233052820)*a(n + 6) + 1015*(5*n + 39)*(n + 8)*(5*n + 41)*(5*n + 37)*(5*n + 38)*a(n + 7), a(0) = 1, a(1) = 2, a(2) = 7, a(3) = 30, a(4) = 141, a(5) = 704, a(6) = 3666}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 20 2026

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(1+x^2)^2))/x)
(PARI) a(n, s=2, t=2, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial(u*(n+1), n-s*k))/(n+1);

Crossrefs

Cf. A000108, A143927.

Cf. A186241, A369440.

Sequence in context: A116363 A186858 A360102 * A371432 A366089 A368936

Adjacent sequences: A369438 A369439 A369440 * A369442 A369443 A369444

Keyword

nonn

Author

Seiichi Manyama, Jan 23 2024

Status

approved