A276753 - OEIS

%I #12 Sep 18 2016 15:48:59

%S 1,5,34,381,8401,334688,27151993,4091831133,1251353635162,

%T 737891198902325,864695662715974585,2033353960345783330704,

%U 9255876152303901497918425,87365856252845525476020365429,1563265999862817889675899566032954,59157049408983740505063226640565220029,4200428372739940183291465697348398947046393,634544126271277747190512917479290795324884131840

%N L.g.f.: Sum_{n>=1} [ Sum_{k>=1} k^(2*n-1) * x^k ]^n / n.

%C L.g.f. equals the logarithm of the g.f. of A276751.

%H Paul D. Hanna, <a href="/A276753/b276753.txt">Table of n, a(n) for n = 1..100</a>

%F L.g.f.: 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 L.g.f.: 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 +...

%e such that A(x) equals the series:

%e 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 This logarithmic series can be written using the Eulerian numbers like so:

%e 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 +...

%e where

%e exp(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 +...+ A276751(n)*x^n +...

%o (PARI) {a(n) = n * polcoeff( sum(m=1, n, sum(k=1, n, k^(2*m-1)*x^k +x*O(x^n))^m/m ), n)}

%o for(n=1, 20, print1(a(n), ", "))

%o (PARI) {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}

%o {a(n) = my(A=1, Oxn=x*O(x^n)); A = 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 ); n * polcoeff(A, n)}

%o for(n=1, 20, print1(a(n), ", "))

%Y Cf. A276751, A276754, A008292.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 17 2016