A215126 E.g.f.: Sum_{n>=0} D^(n*(n-1)/2) (x + x^2)^(n*(n+1)/2) / (n*(n+1)/2)!, where operator D^n = d^n/dx^n.

1, 1, 3, 13, 103, 1071, 16261, 310543, 7712433, 234935473, 8704118611, 385999169421, 20105403167983, 1220933294124943, 85391317006742133, 6817483491016756111, 617003393602674201721, 62822626252518811177953, 7153354574514200634711043, 905989984031687906166904333

Offset

0,3

Comments

Compare to the identity:

exp(x) = Sum_{n>=0} D^(n*(n-1)/2) x^(n*(n+1)/2) / (n*(n+1)/2)!, where operator D^n = d^n/dx^n.

Links

Table of n, a(n) for n=0..19.

Example

 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 103*x^4/4! + 1071*x^5/5! +...
such that, by definition:
A(x) = 1 + (x+x^2) + d/dx (x+x^2)^3/3! + d^3/dx^3 (x+x^2)^6/6! + d^6/dx^6 (x+x^2)^10/10! + d^10/dx^10 (x+x^2)^15/15! +...
Compare to the trivial identity:
exp(x) = 1 + x + d/dx x^3/3! + d^3/dx^3 x^6/6! + d^6/dx^6 x^10/10! + d^10/dx^10 x^15/15! +...

Prog

 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=1+sum(m=1, n, Dx(m*(m-1)/2, (x+x^2+x*O(x^n))^(m*(m+1)/2)/(m*(m+1)/2)!)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A215127.

Sequence in context: A240167 A127004 A068168 * A228148 A098027 A357247

Adjacent sequences: A215123 A215124 A215125 * A215127 A215128 A215129

Keyword

nonn

Author

Paul D. Hanna, Aug 04 2012

Status

approved