A045724 - OEIS

%I #8 Jul 19 2024 21:40:22

%S 1,15,142,1083,7266,44758,259356,1435347,7663898,39761282,201483204,

%T 1001098462,4891910100,23565178380,112118316088,527674017411,

%U 2459747256138,11368724035210,52145629874100,237541552456362

%N Convolution of Catalan numbers A000108 with A020918.

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

%H G. C. Greubel, <a href="/A045724/b045724.txt">Table of n, a(n) for n = 0..1000</a>

%F 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),

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

%t 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 *)

%o (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

%o (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

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

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_