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%I #8 Aug 04 2012 00:39:39
%S 1,1,7,33,223,1753,14391,137137,1382383,14981673,174494983,2135204161,
%T 27643067007,375548195833,5326762882903,78889684038993,
%U 1213984929832591,19377034523034697,320293617185965863,5468629894127442913,96328047496084810783,1747805367475759936281
%N E.g.f.: Sum_{n>=0} d^n/dx^n (x + x^2)^(2*n) / (2*n)!.
%C Compare to the identity:
%C exp(x) = Sum_{n>=0} d^n/dx^n x^(2*n) / (2*n)!.
%e E.g.f.: A(x) = 1 + x + 7*x^2/2! + 33*x^3/3! + 223*x^4/4! + 1753*x^5/5! +...
%e such that, by definition:
%e A(x) = 1 + d/dx (x+x^2)^2/2! + d^2/dx^2 (x+x^2)^4/4! + d^3/dx^3 (x+x^2)^6/6! + d^4/dx^4 (x+x^2)^8/8! + d^5/dx^5 (x+x^2)^10/10! +...
%e Compare to the trivial identity:
%e exp(x) = 1 + d/dx x^2/2! + d^2/dx^2 x^4/4! + d^3/dx^3 x^6/6! + d^4/dx^4 x^8/8! + d^5/dx^5 x^10/10! +...
%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, (x+x^2+x*O(x^n))^(2*m)/(2*m)!)); n!*polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A215128.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 04 2012
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