A045724 Convolution of Catalan numbers A000108 with A020918.

1, 15, 142, 1083, 7266, 44758, 259356, 1435347, 7663898, 39761282, 201483204, 1001098462, 4891910100, 23565178380, 112118316088, 527674017411, 2459747256138, 11368724035210, 52145629874100, 237541552456362

Offset

0,2

Comments

Also convolution of A001700 with A038845; also convolution of A029887 with A000302 (powers of 4); also convolution of A042941 with A000984 (central binomial coefficients).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = binomial(n+4, 3)*A000984(n+4)/(2*A000984(3)) - (n+3)*(n+2)*4^n, where A000984(n) = binomial(2*n, n),

G.f.: c(x)/(1-4*x)^(7/2) = (2 - c(x))/(1-4*x)^4, where c(x) = g.f. for Catalan numbers.

Mathematica

Table[(Binomial[n+5, 4]*CatalanNumber[n+4] -5*4^(n+1)*Binomial[n+3, 2] )/10, {n, 0, 40}] (* G. C. Greubel, Jul 19 2024 *)

Prog

(Magma) [(Binomial(n+5, 4)*Catalan(n+4) -5*4^(n+1)*Binomial(n+3, 2))/10: n in [0..40]]; // G. C. Greubel, Jul 19 2024
(SageMath) [(binomial(n+5, 4)*catalan_number(n+4) - 5*4^(n+1)*binomial(n+3, 2))/10 for n in range(41)] # G. C. Greubel, Jul 19 2024

Crossrefs

Cf. A000108, A000302, A000984, A001700, A020918, A029887, A042941.

Sequence in context: A026859 A096046 A093117 * A241402 A252825 A279921

Adjacent sequences: A045721 A045722 A045723 * A045725 A045726 A045727

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved