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A276751 G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^(2*n-1) * x^k]^n / n ), a power series in x with integer coefficients.

%I A276751 #11 Oct 17 2020 07:39:43
%S A276751 1,1,3,14,111,1813,57846,3941129,515554887,139563384274,
%T A276751 73929755773659,78682910542834037,169524995438153307498,
%U A276751 712160156293232925362965,6241130803695426404771763891,104223975880844169453617144998346,3697419824526049703366356719095712903,247087719554207540966918934263865223952113,35252698554630762784745670915756020139337705854,9472029798481852471047526788494040155248502738148149
%N A276751 G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^(2*n-1) * x^k]^n / n ), a power series in x with integer coefficients.
%C A276751 Conjecture: a(n)^(1/n^2) tends to sqrt(2). - _Vaclav Kotesovec_, Oct 17 2020
%H A276751 Paul D. Hanna, <a href="/A276751/b276751.txt">Table of n, a(n) for n = 0..100</a>
%F A276751 G.f.: exp( Sum_{n>=1} [ Sum_{k=1..2*n-1} A008292(2*n-1,k) * x^k / (1-x)^(2*n) ]^n / n ), where A008292 are the Eulerian numbers.
%e A276751 G.f.: A(x) = 1 + x + 3*x^2 + 14*x^3 + 111*x^4 + 1813*x^5 + 57846*x^6 + 3941129*x^7 + 515554887*x^8 + 139563384274*x^9 + 73929755773659*x^10 +...
%e A276751 log(A(x)) = x + 5*x^2/2 + 34*x^3/3 + 381*x^4/4 + 8401*x^5/5 + 334688*x^6/6 + 27151993*x^7/7 + 4091831133*x^8/8 + 1251353635162*x^9/9 + 737891198902325*x^10/10 + 864695662715974585*x^11/11 + 2033353960345783330704*x^12/12 +...
%e A276751 The logarithm of g.f. A(x) equals the series:
%e A276751 log(A(x)) = Sum_{n>=1} (x + 2^(2*n-1)*x^2 + 3^(2*n-1)*x^3 +...+ k^(2*n-1)*x^k +...)^n/n.
%e A276751 This logarithmic series can be written using the Eulerian numbers like so:
%e A276751 log(A(x)) = x/(1-x)^2 + (x + 4*x^2 + x^3)^2/(1-x)^8/2 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^3/(1-x)^18/3 + (x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7)^4/(1-x)^32/4 + (x + 502*x^2 + 14608*x^3 + 88234*x^4 + 156190*x^5 + 88234*x^6 + 14608*x^7 + 502*x^8 + x^9)^5/(1-x)^50/5 + (x + 2036*x^2 + 152637*x^3 + 2203488*x^4 + 9738114*x^5 + 15724248*x^6 + 9738114*x^7 + 2203488*x^8 + 152637*x^9 + 2036*x^10 + x^11)^6/(1-x)^72/6 +...+ [ Sum_{k=1..2*n-1} A008292(2*n-1,k) * x^k ]^n / (1-x)^(2*n^2) /n +...
%o A276751 (PARI) {a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n+1, k^(2*m-1) * x^k +x*O(x^n))^m / m ) ), n)}
%o A276751 for(n=0, 20, print1(a(n), ", "))
%o A276751 (PARI) {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
%o A276751 {a(n) = my(A=1, Oxn=x*O(x^n)); A = exp( sum(m=1, n+1, sum(k=1, 2*m-1, A008292(2*m-1, k)*x^k/(1-x +Oxn)^(2*m) )^m / m ) ); polcoeff(A, n)}
%o A276751 for(n=0, 20, print1(a(n), ", "))
%Y A276751 Cf. A156170, A276752.
%K A276751 nonn
%O A276751 0,3
%A A276751 _Paul D. Hanna_, Sep 16 2016