A249015 A binomial convolution.

1, 1, 5, 17, 69, 339, 1677, 9321, 55137, 343659, 2285289, 15910857, 116120781, 886308147, 7033465989, 58008074409, 495792941337, 4381170220251, 39980186877537, 376025841184329, 3640077999981189, 36224841818288547, 370112212444620861, 3878334404076375657

Offset: 0

Author

Emanuele Munarini, Oct 20 2014

Keywords

• nonn

Crossrefs

Cf. A049425, A252284.

Programs

Mathematica

b[n_] := Sum[(n!/k!)Sum[Binomial[k,i]Binomial[k-i+2,n-2i-k]/3^i,{i,0,k}],{k,0,n}]
c[n_] := Sum[(n!/k!)(-1)^k Sum[Binomial[k,i]Binomial[k-i,n-2i-k]/3^i,{i,0,k}],{k,0,n}]
Table[If[n==0,1,0]+Sum[Binomial[n,k]b[k]c[n-k-1],{k,0,n-1}],{n,0,40}]

Maxima

b(n) := sum((n!/k!)*sum(binomial(k,i)*binomial(k-i+2,n-2*i-k)/3^i,i,0,k),k,0,n);
c(n) := sum((n!/k!)*(-1)^k*sum(binomial(k,i)*binomial(k-i,n-2*i-k)/3^i,i,0,k),k,0,n);
makelist((if n=0 then 1 else 0)+sum(binomial(n,k)*b(k)*c(n-k-1),k,0,n-1),n,0,20);

Formula

a(n) = 0^0 + Sum_{k=0..n-1} binomial(n,k)*b(k)*c(n-k-1),

where the numbers b(n) = A049425(n+1) have e.g.f. (1+t)^2*exp(t+t^2+t^3/3)

and the numbers c(n) have e.g.f. exp(-(t+t^2+t^3/3)).

D-finite with recurrence: a(n+4) + n*a(n+3) - 3*(n+3)*a(n+2) - 3*(n+3)*(n+2)*a(n+1) - (n+3)*(n+2)*(n+1)*a(n) = 0.

E.g.f.: A(t) = 1+(1+t)^2*exp(t+t^2+t^3/3)*Integral_{u=0..t} exp(-(u+u^2+u^3/3)) du.

Differential equation for the e.g.f.: (1+t)*A''(t) - (2+3*t+3*t^2+t^3)*A'(t) - 3*(1+t)^2*A(t) = 0.