A259209 - OEIS

%I #19 Aug 27 2023 08:12:50

%S 1,2,3,16,5,366,7,10088,60489,302410,11,89812812,13,363242894,

%T 108972864015,886312627216,17,178478870169618,19,101401086923136020,

%U 354798209525760021,1548722343168022,23,13787827750211997081624,129260083694424883200025,5051650697533440026

%N E.g.f.: Sum_{n>=1} x^(n^2) * exp(x^n) / n!.

%F E.g.f.: -exp(1) + Sum_{n>=0} (1 + x^n)^n / n!.

%F a(n) = Sum_{d|n} binomial(d, n/d) * n!/d! for n>=1.

%e E.g.f.: A(x) = x + 2*x^2/2! + 3*x^3/3! + 16*x^4/4! + 5*x^5/5! + 366*x^6/6! +...

%e where

%e A(x) = x*exp(x) + x^4*exp(x^2)/2! + x^9*exp(x^3)/3! + x^16*exp(x^4)/4! + x^25*exp(x^5)/5! + x^36*exp(x^6)/6! +...

%e also

%e A(x) = -exp(1) + 1 + (1+x) + (1+x^2)^2/2! + (1+x^3)^3/3! + (1+x^4)^4/4! + (1+x^5)^5/5! + (1+x^6)^6/6! +...

%e Particular values.

%e A(1) = exp(2) - exp(1).

%e A(-1) = cosh(2) - exp(1).

%e A(1/2) = 0.8648559700938957468696599588156983897723576531...

%e A(1/2) = exp(1/2)/2 + exp(1/2^2)/(2!*2^4) + exp(1/2^3)/(3!*2^9) + exp(1/2^4)/(4!*2^16) + exp(1/2^5)/(5!*2^25) +...

%e A(1/2) = -exp(1) + 1 + (1+1/2) + (1+1/2^2)^2/2! + (1+1/2^3)^3/3! + (1+1/2^4)^4/4! + (1+1/2^5)^5/5! + (1+1/2^6)^6/6! +...

%t Table[Sum[Binomial[d, n/d]*n!/d!, {d, Divisors[n]}], {n, 1, 30}] (* _Vaclav Kotesovec_, Oct 20 2020 *)

%o (PARI) {a(n) = my(A=1); A = sum(m=1, n, x^(m^2) * exp(x^m +x*O(x^n)) / m!); n!*polcoeff(A, n)}

%o for(n=1, 30, print1(a(n), ", "))

%o (PARI) {a(n) = my(A=1); A = -exp(1) + sum(m=0, n, (1 + x^m +x*O(x^n))^m/m!); if(n==0,0, n!*polcoeff(A, n))}

%o for(n=1, 30, print1(a(n), ", "))

%o (PARI) {a(n) = if(n<1,0, sumdiv(n,d, binomial(d, n/d) * n!/d! ) )}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A259223, A259208, A265943, A265270, A266211.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jun 21 2015