0,2

More generally, for fixed integer q, G(x,q) = Sum_{n>=0} exp(-n * q^n * x) * [ Sum_{k>=1} k^n * q^(n*k) * x^k / k! ]^n is an integer series such that G(x,q) = Sum_{n>=0} q^(n^2) * [ Sum_{k=1..n} S2(n,k) * q^(n*k-n) * x^k ]^n.

G.f.: Sum_{n>=0} [ Sum_{k=1..n} S2(n,k) * 2^(n*k) * x^k ]^n, where S2(n,k) = A008277(n,k) are the Stirling numbers of the second kind.

G.f.: A(x) = 1 + 2*x + 16*x^2 + 640*x^3 + 102656*x^4 + 63897600*x^5 + 154597064704*x^6 + 1463095704682496*x^7 +...

such that

A(x) = Sum_{n>=0} exp(-n*2^n*x) * (2^n*x + 2^n*2^(2*n)*x^2/2! + 3^n*2^(3*n)*x^3/3! +...+ k^n*2^(k*n)*x^k/k! +...)^n.

Explicitly,

A(x) = 1 + exp(-2*x) * (2*x + 2*2^2*x^2/2! + 3*2^3*x^3/3! + 4*2^4*x^4/4! +...) +

exp(-2*2^2*x) * (2^2*x + 4*2^4*x^2/2! + 9*2^6*x^3/3! + 16*2^8*x^4/4! +...)^2 +

exp(-3*2^3*x) * (2^3*x + 8*2^6*x^2/2! + 27*2^9*x^3/3! + 64*2^12*x^4/4! +...)^3 +

exp(-4*2^4*x) * (2^4*x + 16*2^8*x^2/2! + 81*2^12*x^3/3! + 256*2^16*x^4/4! +...)^4 +

exp(-5*2^5*x) * (2^5*x + 32*2^10*x^2/2! + 243*2^15*x^3/3! + 1024*2^20*x^4/4! +...)^5 +...

The g.f. can be written using the Stirling2 numbers like so:

A(x) = 1 + 2*x + (2^2*x + 2^4*x^2)^2 + (2^3*x + 3*2^6*x^2 + 2^9*x^3)^3 + (2^4*x + 7*2^8*x^2 + 6*2^12*x^3 + 2^16*x^4)^4 + (2^5*x + 15*2^10*x^2 + 25*2^15*x^3 + 10*2^20*x^4 + 2^25*x^5)^5 + (2^6*x + 31*2^12*x^2 + 90*2^18*x^3 + 65*2^24*x^4 + 15*2^30*x^5 + 2^36*x^6)^6 + (2^7*x + 63*2^14*x^2 + 301*2^21*x^3 + 350*2^28*x^4 + 140*2^35*x^5 + 21*2^42*x^6 + 2^49*x^7)^7 +...+ [ Sum_{k=1..n} S2(n,k) * 2^(n*k) * x^k ]^n +...

(PARI) {a(n) = my(A=1); A = sum(m=0, n+1, exp(-m*2^m*x +x*O(x^n)) * sum(k=1, n+1, 2^(m*k)*k^m*x^k/k! +x*O(x^n))^m ); polcoeff(A, n)}

for(n=0, 15, print1(a(n), ", "))

(PARI) {a(n) = my(A = sum(m=0, n, sum(k=1, m, stirling(m, k, 2)*2^(m*k)*x^k +x*O(x^n) )^m )); polcoeff(A, n)}

for(n=0, 15, print1(a(n), ", "))

Cf. A276746, A277037, A168407.

Sequence in context: A369674 A012757 A012464 * A289202 A279324 A128294

Adjacent sequences: A277033 A277034 A277035 * A277037 A277038 A277039

Paul D. Hanna, Oct 08 2016

approved