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%I #14 Jan 30 2022 11:40:41
%S 1,1,2,14,168,3147,90563,3561231,185790622,12599020184,1071164190670,
%T 111813313594259,14140296360430353,2132273568722682621,
%U 378197030144360862958,78127192632748956075174,18627308660113953164384812,5081218748742336002185874439,1574128413278644602881499193579
%N O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - n^2*k*x).
%H Seiichi Manyama, <a href="/A229257/b229257.txt">Table of n, a(n) for n = 0..242</a>
%F a(n) = Sum_{k=0..n} (k^2)^(n-k) * Stirling2(n, k).
%F E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n / (n! * n^(2*n)).
%e O.g.f.: A(x) = 1 + x + 2*x^2 + 14*x^3 + 168*x^4 + 3147*x^5 + 90563*x^6 +...
%e where
%e A(x) = 1 + x/(1-x) + x^2/((1-2^2*1*x)*(1-2^2*2*x)) + x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
%e Exponential Generating Function.
%e E.g.f.: E(x) = 1 + x + 2*x^2/2! + 14*x^3/3! + 168*x^4/4! + 3147*x^5/5! +...
%e where
%e E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/(2!*4^2) + (exp(9*x)-1)^3/(3!*9^3) + (exp(16*x)-1)^4/(4!*16^4) + (exp(25*x)-1)^5/(5!*25^5) +...
%t Flatten[{1,Table[Sum[(k^2)^(n-k) * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, May 08 2014 *)
%o (PARI) {a(n)=polcoeff(sum(m=0,n,x^m/prod(k=1,m,1-m^2*k*x +x*O(x^n))),n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n,(exp(m^2*x+x*O(x^n))-1)^m/(m!*m^(2*m))),n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=sum(k=0, n, (k^2)^(n-k) * stirling(n, k, 2))}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A229258, A229259, A229260, A229261; A229233, A229234, A220181, A122399.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 17 2013
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