A045724 - OEIS

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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

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

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