A215127 E.g.f.: Sum_{n>=0} D^(n^2-n) (x + x^2)^(n^2) / (n^2)!, where operator D^n = d^n/dx^n.

1, 1, 3, 21, 271, 5073, 149931, 5629933, 287996871, 18574155561, 1472489126563, 143431714523781, 16629096827674623, 2271941249486405761, 362871752515734614811, 66782754543872231839773, 14054632818067589280068791, 3359850327080126215443462873

Offset

0,3

Comments

Compare to the identity:

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

Links

Table of n, a(n) for n=0..17.

Example

 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 21*x^3/3! + 271*x^4/4! + 5073*x^5/5! +...
such that, by definition:
A(x) = 1 + (x+x^2) + d^2/dx^2 (x+x^2)^4/4! + d^6/dx^6 (x+x^2)^9/9! + d^12/dx^12 (x+x^2)^16/16! + d^20/dx^20 (x+x^2)^25/25! +...
Compare to the trivial identity:
exp(x) = 1 + x + d^2/dx^2 x^4/4! + d^6/dx^6 x^9/9! + d^12/dx^12 x^16/16! + d^20/dx^20 x^25/25! +...

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^2-m, (x+x^2+x*O(x^n))^(m^2)/(m^2)!)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A215126.

Sequence in context: A098278 A269938 A277454 * A227820 A336809 A066206

Adjacent sequences: A215124 A215125 A215126 * A215128 A215129 A215130

Keyword

nonn

Author

Paul D. Hanna, Aug 04 2012

Status

approved