A369617 - OEIS

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A369617

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

1, 4, 22, 146, 1079, 8525, 70468, 601816, 5268241, 47019566, 426250277, 3914020148, 36328457669, 340278596273, 3212416054283, 30534649412247, 291981031204917, 2806832429353512, 27109863184695640, 262951127248539898, 2560229132085602215

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..984

Index entries for reversions of series

FORMULA

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

D-finite with recurrence 3*(3*n+2)*(3*n+1)*(n+1)*a(n) +4*(-91*n^3 -32*n^2 +n+2)*a(n-1) +2*(n-1)*(465*n^2 -337*n+86)*a(n-2) -4*(n-1)*(n-2) *(219*n-187)*a(n-3) +283*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Jan 28 2024

MAPLE

A369617 := proc(n)

add(binomial(n+1, k) * binomial(4*n-4*k+2, n-k), k=0..n) ;

%/(n+1) ;

end proc;

seq(A369617(n), n=0..70) ; # R. J. Mathar, Jan 28 2024

PROG

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)^3+x))/x)

(PARI) a(n) = sum(k=0, n, binomial(n+1, k)*binomial(4*n-4*k+2, n-k))/(n+1);

CROSSREFS

Cf. A007317, A369616, A370844.

Cf. A006632, A274735.