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

0,3

Links

Table of n, a(n) for n=0..23.

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.

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}]

Prog

(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);

Crossrefs

Cf. A049425, A252284.

Sequence in context: A273763 A149707 A214003 * A273852 A146511 A128887

Adjacent sequences: A249012 A249013 A249014 * A249016 A249017 A249018

Keyword

nonn

Author

Emanuele Munarini, Oct 20 2014

Status

approved