A360349 - OEIS

%I #10 Feb 13 2023 03:40:43

%S 1,1,5,38,391,5077,79535,1458264,30621237,724555611,19076629520,

%T 553236991215,17525729241605,602215048797900,22312035980459259,

%U 886733059906749795,37631474149766344476,1698581174869953607957,81257725943229600518977,4106922637708383448243974

%N G.f. A(x) = exp( Sum_{k>=1} A360348(k) * x^k/k ), where A360348(k) = [y^k*x^k/k] log( Sum_{m>=0} (1 + m*y + y^2)^m * x^m ) for k >= 1.

%C Related series: M(x) = exp( Sum_{k>=1} A002426(k) * x^k/k ), where M(x) = 1 + x*M(x) + x^2*M(x)^2 is the Motzkin function (A001006) and A002426(k) = [y^k*x^k/k] log( Sum_{m>=0} (1 + y + y^2)^m * x^m ) for k >= 1.

%H Paul D. Hanna, <a href="/A360349/b360349.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) ~ BesselI(0, 2) * n^n. - _Vaclav Kotesovec_, Feb 12 2023

%e G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 391*x^4 + 5077*x^5 + 79535*x^6 + 1458264*x^7 + 30621237*x^8 + 724555611*x^9 + ...

%e such that

%e log(A(x)) = x + 9*x^2/2 + 100*x^3/3 + 1381*x^4/4 + 22771*x^5/5 + 435138*x^6/6 + 9442049*x^7/7 + 229265109*x^8/8 + ... + A360348(n)*x^n/n + ...

%e where A360348(n) equals the coefficient of y^n*x^n/n in the logarithmic series:

%e log( Sum_{m>=0} (1 + m*y + y^2)^m * x^m ) = (y^2 + y + 1)*x + (y^4 + 6*y^3 + 9*y^2 + 6*y + 1)*x^2/2 + (y^6 + 15*y^5 + 63*y^4 + 100*y^3 + 63*y^2 + 15*y + 1)*x^3/3 + (y^8 + 28*y^7 + 242*y^6 + 872*y^5 + 1381*y^4 + 872*y^3 + 242*y^2 + 28*y + 1)*x^4/4 + (y^10 + 45*y^9 + 665*y^8 + 4430*y^7 + 14545*y^6 + 22771*y^5 + 14545*y^4 + 4430*y^3 + 665*y^2 + 45*y + 1)*x^5/5 + ...

%o (PARI) {A360348(n) = n * polcoeff( polcoeff( log( sum(m=0, n+1, (1 + m*y + y^2)^m *x^m ) +x*O(x^n) ), n, x), n, y)}

%o {a(n) = polcoeff( exp( sum(m=1,n, A360348(m)*x^m/m ) +x*O(x^n)),n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A360348, A360239, A186925, A001006, A002426.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 12 2023