A379081 Expansion of (1/x) * Series_Reversion( x * (1/(1 + x) - x^2)^3 ).

1, 3, 15, 94, 657, 4905, 38299, 308928, 2554092, 21528728, 184318944, 1598427531, 14011401996, 123946608699, 1105090991634, 9920335032821, 89589290332200, 813367589142888, 7419376746340780, 67965042988027335, 624971955439306953, 5766825797557702751, 53380176096582823851

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

G.f. A(x) satisfies:

(1) A(x) = exp( Sum_{k>=1} A379086(k) * x^k/k ).

(2) A(x) = ( (1 + x*A(x)) * (1 + x^2*A(x)^(7/3)) )^3.

(3) A(x) = B(x)^3 where B(x) is the g.f. of A379088.

a(n) = (1/(n+1)) * [x^n] 1/( 1/(1 + x) - x^2 )^(3*(n+1)).

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

Prog

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

Crossrefs

Cf. A181734, A379080.

Cf. A379086, A379088.

Sequence in context: A128240 A369270 A369301 * A368964 A274734 A378951

Adjacent sequences: A379078 A379079 A379080 * A379082 A379083 A379084

Keyword

nonn

Author

Seiichi Manyama, Dec 15 2024

Status

approved