cp's OEIS Frontend

A360238 a(n) = [y^n*x^n/n] log( Sum_{m>=0} (m + y)^(2*m) * x^m ) for n >= 1.

%I A360238 #20 Feb 13 2023 03:42:24
%S A360238 2,42,1376,60934,3377252,224036904,17282039280,1519096411230,
%T A360238 149867251224092,16398595767212452,1971137737765484444,
%U A360238 258215735255164847944,36617351885600586385222,5588967440618883091216208,913592455995572681826313856,159241707066923571547572653630
%N A360238 a(n) = [y^n*x^n/n] log( Sum_{m>=0} (m + y)^(2*m) * x^m ) for n >= 1.
%C A360238 Related sequence: A000984(n) = binomial(2*n,n) = [y^n*x^n/n] log( Sum_{m>=0} (1 + y)^(2*m) * x^m ) for n >= 1.
%H A360238 Paul D. Hanna, <a href="/A360238/b360238.txt">Table of n, a(n) for n = 1..200</a>
%F A360238 a(n) ~ (1 - exp(-1)/4) * 2^(2*n) * n^(n + 1/2) / sqrt(Pi). - _Vaclav Kotesovec_, Feb 12 2023
%e A360238 L.g.f.: A(x) = 2*x + 42*x^2/2 + 1376*x^3/3 + 60934*x^4/4 + 3377252*x^5/5 + 224036904*x^6/6 + 17282039280*x^7/7 + 1519096411230*x^8/8 + ...
%e A360238 a(n) equals the coefficient of y^n*x^n/n in the logarithmic series:
%e A360238 log( Sum_{m>=0} (m + y)^(2*m) * x^m ) = (y^2 + 2*y + 1)*x + (y^4 + 12*y^3 + 42*y^2 + 60*y + 31)*x^2/2 + (y^6 + 30*y^5 + 297*y^4 + 1376*y^3 + 3348*y^2 + 4188*y + 2140)*x^3/3 + (y^8 + 56*y^7 + 1100*y^6 + 10792*y^5 + 60934*y^4 + 209464*y^3 + 436692*y^2 + 510952*y + 258779)*x^4/4 + (y^10 + 90*y^9 + 2945*y^8 + 49960*y^7 + 510160*y^6 + 3377252*y^5 + 14971780*y^4 + 44457000*y^3 + 85336175*y^2 + 96141170*y + 48446971)*x^5/5 + (y^12 + 132*y^11 + 6486*y^10 + 169236*y^9 + 2730921*y^8 + 29547696*y^7 + 224036904*y^6 + 1214958240*y^5 + 4717830978*y^4 + 12868488144*y^3 + 23497266672*y^2 + 25858665696*y + 12994749280)*x^6/6 + ...
%e A360238 Exponentiation yields the g.f. of A360239:
%e A360238 exp(A(x)) = 1 + 2*x + 23*x^2 + 502*x^3 + 16414*x^4 + 716936*x^5 + 39167817*x^6 + 2567058766*x^7 + 196159319943*x^8 + ... + A360239(n)*x^n + ...
%o A360238 (PARI) {a(n) = n * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(2*m) *x^m ) +x*O(x^n) ), n, x), n, y)}
%o A360238 for(n=0,20,print1(a(n),", "))
%Y A360238 Cf. A360239, A360348, A266526, A000984, A059304, A098658.
%K A360238 nonn
%O A360238 1,1
%A A360238 _Paul D. Hanna_, Feb 11 2023