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1, -2, 13, -248, 12526, -1568368, 466802541, -321051272000, 500039245711658, -1737306124802148608, 13308275973878544047746, -222688458529322994469714944, 8077969377058605224894763722940, -631225844487016628864332741755017216, 105685667980009079816649620724931814050429, -37739472986084796740536023316481369102235181056

Paul D. Hanna, <a href="/A276744/b276744.txt">Table of n, a(n) for n = 1..120</a>

Cf. A276743, A276745, A008292.

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Cf. A276743, A008292.

allocated for Paul D. Hanna

G.f.: x = Sum_{n>=1} a(n) * [ Sum_{k>=1} k^n * x^k ]^n.

1, -2, 13, -248, 12526, -1568368, 466802541, -321051272000, 500039245711658, -1737306124802148608, 13308275973878544047746, -222688458529322994469714944, 8077969377058605224894763722940, -631225844487016628864332741755017216, 105685667980009079816649620724931814050429, -37739472986084796740536023316481369102235181056

1,2

G.f.: x = Sum_{n>=1} a(n) * [ Sum_{k=1..n} A008292(n,k) * x^k ]^n / (1-x)^(n*(n+1)), where A008292 are the Eulerian numbers.

G.f.: x = Sum_{n>=1} a(n) * (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n.

The g.f. can be written using the Eulerian numbers like so:

x = x/(1-x)^2 - 2*(x + x^2)^2/(1-x)^6 + 13*(x + 4*x^2 + x^3)^3/(1-x)^12 - 248*(x + 11*x^2 + 11*x^3 + x^4)^4/(1-x)^20 + 12526*(x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5/(1-x)^30 - 1568368*(x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6/(1-x)^42 +...+ a(n)*[ Sum_{k=1..n} A008292(n,k) * x^k ]^n / (1-x)^(n*(n+1)) +...

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);

A[#A] = -Vec(sum(m=1, #A, A[m]*sum(k=1, #A+1, k^m * x^k +x*O(x^#A))^m))[#A] ); A[n]}

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

Cf. A008292.

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Paul D. Hanna, Sep 29 2016

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allocated for Paul D. Hanna

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