A215127 - OEIS

%I #3 Aug 04 2012 00:57:18

%S 1,1,3,21,271,5073,149931,5629933,287996871,18574155561,1472489126563,

%T 143431714523781,16629096827674623,2271941249486405761,

%U 362871752515734614811,66782754543872231839773,14054632818067589280068791,3359850327080126215443462873

%N  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.

%C  Compare to the identity:

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

%e  E.g.f.: A(x) = 1 + x + 3*x^2/2! + 21*x^3/3! + 271*x^4/4! + 5073*x^5/5! +...

%e such that, by definition:

%e 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! +...

%e Compare to the trivial identity:

%e 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! +...

%o  (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {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)}

%o for(n=0, 25, print1(a(n), ", "))

%Y  Cf. A215126.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 04 2012