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A249015 A binomial convolution.

%I A249015 #19 Aug 11 2021 09:31:15
%S A249015 1,1,5,17,69,339,1677,9321,55137,343659,2285289,15910857,116120781,
%T A249015 886308147,7033465989,58008074409,495792941337,4381170220251,
%U A249015 39980186877537,376025841184329,3640077999981189,36224841818288547,370112212444620861,3878334404076375657
%N A249015 A binomial convolution.
%F A249015 a(n) = 0^0 + Sum_{k=0..n-1} binomial(n,k)*b(k)*c(n-k-1),
%F A249015 where the numbers b(n) = A049425(n+1) have e.g.f. (1+t)^2*exp(t+t^2+t^3/3)
%F A249015 and the numbers c(n) have e.g.f. exp(-(t+t^2+t^3/3)).
%F A249015 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.
%F A249015 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.
%F A249015 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.
%t A249015 b[n_] := Sum[(n!/k!)Sum[Binomial[k,i]Binomial[k-i+2,n-2i-k]/3^i,{i,0,k}],{k,0,n}]
%t A249015 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}]
%t A249015 Table[If[n==0,1,0]+Sum[Binomial[n,k]b[k]c[n-k-1],{k,0,n-1}],{n,0,40}]
%o A249015 (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);
%o A249015 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);
%o A249015 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);
%Y A249015 Cf. A049425, A252284.
%K A249015 nonn
%O A249015 0,3
%A A249015 _Emanuele Munarini_, Oct 20 2014