A249014 A double binomial sum.

1, 3, 15, 105, 933, 9951, 123123, 1727685, 27050985, 466795323, 8791179831, 179262508833, 3931730998605, 92237649141015, 2303515063987803, 60987344488950141, 1705641174191204433, 50228924171214633075, 1553143164997199612895

Offset: 0

Author

Emanuele Munarini, Oct 20 2014

Keywords

• nonn

Crossrefs

Cf. A049425.

Programs

Mathematica

Table[Sum[n!/k!Sum[Binomial[k,i]Binomial[n+k-i+1,2k+i+1]/3^i,{i,0,k}],{k,0,n}],{n,0,60}]

Maxima

makelist(sum(n!/k!*sum(binomial(k,i)*binomial(n+k-i+1,2*k+i+1)/3^i,i,0,k),k,0,n),n,0,12);

Formula

E.g.f.: (1/(1-t)^2)*exp((3*t-3*t^2+t^3)/(3*(1-t)^3)).

a(n) = sum(n!/k!*sum(bin(k,i)*bin(n+k-i+1,2*k+i+1)/3^i,i=0..k),k=0..n).

a(n) = sum(Lah(n,k)*h(k),k=0..n), where Lah(n,k) are the Lah numbers and the numbers h(n) are defined by the e.g.f. h(x) = (1+t)^2*exp(t+t^2+t^3/3) (essentially sequence A049425).

a(n) = sum(Lah(n+1,k+1)*h(k),k=0..n), where Lah(n,k) are the Lah numbers and the numbers h(n) are defined by the e.g.f. h(x) = exp(t+t^2+t^3/3) (sequence A049425).

a(n) = sum(bin(n,k)*(n!/k!)*h(k),k=0..n), where the numbers h(n) are defined by the e.g.f. h(x) = (1+t)*exp(t+t^2+t^3/3).

Recurrence: a(n+4)-(4*n+15)*a(n+3)+6*(n+3)^2*a(n+2)-2*(n+3)*(n+2)*(2n+5)*a(n+1)+(n+3)*(n+2)^2*(n+1)*a(n)=0.