A274735 G.f. satisfies A(x) = (1 + x*A(x))^3 * (1 + x*A(x)^2).

1, 4, 26, 210, 1901, 18445, 187524, 1971672, 21263360, 233907762, 2614446624, 29607343948, 338977591904, 3917185497535, 45629006313280, 535199773167207, 6315789123860388, 74932400322972992, 893276792585933870, 10694510040508714014, 128531711285410216883, 1550159476645634696615, 18755239991772817629972, 227577929298568261967650, 2768820313297861609739979

Offset

0,2

Comments

More generally, if G(x) satisfies

G(x) = (1 + a*x*G(x))^m * (1 + b*x*G(x)^2), then

G(x) = (1/x) * Series_Reversion( x * (1 - b*x*(1 + a*x)^m) / (1 + a*x)^m ).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..903

Formula

G.f.: (1/x) * Series_Reversion( x * (1 - x*(1+x)^3) / (1+x)^3 ).

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(3*n+3*k+3,n-k). - Seiichi Manyama, Jan 27 2024

Example

G.f.: A(x) = 1 + 4*x + 26*x^2 + 210*x^3 + 1901*x^4 + 18445*x^5 + 187524*x^6 + 1971672*x^7 + 21263360*x^8 +...

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = (1 + x*A)^3 * (1 + x*A^2) + x*O(x^n) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x*(1+x)^3)/(1+x +x^2*O(x^n) )^3 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A007863, A274734.

Cf. A274379, A369600.

Sequence in context: A291533 A363363 A239295 * A355379 A349719 A283828

Adjacent sequences: A274732 A274733 A274734 * A274736 A274737 A274738

Keyword

nonn

Author

Paul D. Hanna, Aug 02 2016

Status

approved