A369619 -id:A369619

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A369618

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

+10
2

1, 1, 3, 8, 26, 87, 308, 1122, 4196, 15995, 61953, 243091, 964307, 3860820, 15581231, 63318119, 258874412, 1064093055, 4394851631, 18229191263, 75904152261, 317161817092, 1329465389854, 5589012748902, 23558679647629, 99548061405442, 421598837336178

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..26.

Index entries for reversions of series

FORMULA

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

D-finite with recurrence +(n+2)*(n+1)*a(n) -(n+1)*(13*n-4)*a(n-1) +2*(26*n^2-36*n+1)*a(n-2) +(-61*n^2+219*n-176)*a(n-3) +2*(-26*n^2+78*n-7)*a(n-4) +(235*n-497)*(n-4)*a(n-5) -93*(n-4)*(n-5)*a(n-6)=0. - R. J. Mathar, Jan 28 2024

PROG

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)+x^2))/x)

(PARI) a(n) = sum(k=0, n\2, binomial(n+1, k)*binomial(2*n-3*k, n-2*k))/(n+1);

CROSSREFS

Cf. A369619, A369620.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Jan 27 2024

STATUS

approved

A369620

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

+10
2

1, 3, 16, 100, 692, 5099, 39240, 311700, 2536490, 21037102, 177176745, 1511211409, 13027296723, 113319727772, 993422328313, 8768003882546, 77848008692270, 694828468698510, 6230785015298952, 56109079416527835, 507188912618646021, 4600432953729579585