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%I #3 Jun 05 2019 21:22:23
%S 1,2,7,4,77,6,2527,8,33609,725770,907211,12,946028173,14,968647695,
%T 653837184016,17473020364817,18,935267389056019,20,
%U 1723379337808128021,1703031405723648022,3716933623603223,24,124520749358323872153625,3877802510832746496000026,11787184960911360027,10802449851605508096000028,16938242101749730412851200029,30,51981534567681070815925862400031
%N E.g.f.: Sum_{n>=0} (n+1) * (1 + x^n)^n * x^n/n!
%F E.g.f.: Sum_{n>=0} (n+1) * (1 + x^n)^n * x^n/n!.
%F E.g.f.: Sum_{n>=1} (n + x^n) * exp(x^n) * x^(n*(n-1))/(n-1)!.
%F a(n) = Sum_{d|n} (d + 1) * binomial(d, n/d - 1) * n!/d! for n>0, with a(0) = 1.
%e E.g.f.: A(x) = 1 + 2*x + 7*x^2 + 4*x^3 + 77*x^4 + 6*x^5 + 2527*x^6 + 8*x^7 + 33609*x^8 + 725770*x^9 + 907211*x^10 + 12*x^11 + 946028173*x^12 + 14*x^13 + ...
%e such that
%e A(x) = 1 + 2*(1+x)*x + 3*(1+x^2)^2*x^2/2! + 4*(1+x^3)^3*x^3/3! + 5*(1+x^4)^4*x^4/4! + 6*(1+x^5)^5*x^5/5! + 7*(1+x^6)^6*x^6/6! + 8*(1+x^7)^7*x^7/7! + ...
%e also
%e A(x) = (1 + x)*exp(x) + (2 + x^2)*exp(x^2)*x^2 + (3 + x^3)*exp(x^3)*x^6/2! + (4 + x^4)*exp(x^4)*x^12/3! + (5 + x^5)*exp(x^5)*x^20/4! + (6 + x^6)*exp(x^6)*x^30/5! + (7 + x^7)*exp(x^7)*x^42/6! + ...
%o (PARI) {a(n) = if(n==0,1, sumdiv(n,d, (d + 1) * binomial(d, n/d - 1) * n!/d! ) )}
%o for(n=0,30, print1(a(n),", "))
%o (PARI) /* E.g.f.: Sum_{n>=0} (n+1) * (1 + x^n)^n * x^n/n! */
%o {a(n) = my(A = sum(m=0,n, (m+1) * (x^m + 1 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A,n)}
%o for(n=0,30, print1(a(n),", "))
%o (PARI) /* E.g.f.: Sum_{n>=1} (n + x^n) * exp(x^n) * x^(n*(n-1))/(n-1)! */
%o {a(n) = my(A = sum(m=1,sqrtint(2*n+1), (m + x^m) * exp(x^m +x*O(x^n)) * x^(m*(m-1))/(m-1)! )); n!*polcoeff(A,n)}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A259208.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 05 2019