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

Author

Wolfdieter Lang

Keywords

• easy
• nonn

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

Crossrefs

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

Programs

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

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 *)

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

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.