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A249477 E.g.f.: exp(4)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} 4^n/Product_{k=1..n} (k - x^k).

%I A249477 #6 Feb 07 2025 14:27:09
%S A249477 1,1,7,47,360,2884,26068,250140,2659544,30188024,373401768,4911407656,
%T A249477 69701336160,1046114985408,16770977757888,283455401409920,
%U A249477 5076208319560320,95434083840830080,1890657361059194240,39170792604756397440,850920224456551054336,19275340855527901297152
%N A249477 E.g.f.: exp(4)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} 4^n/Product_{k=1..n} (k - x^k).
%C A249477 The function P(x) = Product_{n>=1} 1/(1 - x^n/n) equals the e.g.f. of A007841, the number of factorizations of permutations of n letters into cycles in nondecreasing length order.
%e A249477 E.g.f.: A(x) = 1 + x + 7*x^2/2! + 47*x^3/3! + 360*x^4/4! + 2884*x^5/5! +...
%e A249477 such that A(x) = exp(4)*P(x) - Q(x), where
%e A249477 P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
%e A249477 Q(x) = Sum_{n>=1} 4^n / Product_{k=1..n} (k - x^k).
%e A249477 More explicitly,
%e A249477 P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...);
%e A249477 Q(x) = 4/(1-x) + 4^2/((1-x)*(2-x^2)) + 4^3/((1-x)*(2-x^2)*(3-x^3)) + 4^4/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 4^5/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +...
%e A249477 We can illustrate the initial terms a(n) in the following manner.
%e A249477 The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
%e A249477 q(0) = 53.59815003314423907811...
%e A249477 q(1) = 53.59815003314423907811...
%e A249477 q(2) = 156.7944500994327172343...
%e A249477 q(3) = 553.5796503645866298592...
%e A249477 q(4) = 2697.496401856077388374...
%e A249477 q(5) = 14805.80061073873346130...
%e A249477 q(6) = 100818.1006770272116175...
%e A249477 q(7) = 750753.2864076001907799...
%e A249477 q(8) = 6488048.449153118392102...
%e A249477 q(9) = 61223693.06709220629587...
%e A249477 and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n) begin:
%e A249477 A007841 = [1, 1, 3, 11, 56, 324, 2324, 18332, 167544, ...];
%e A249477 from which we can generate this sequence like so:
%e A249477 a(0) = exp(4)*1 - q(0) = 1;
%e A249477 a(1) = exp(4)*1 - q(1) = 1;
%e A249477 a(2) = exp(4)*3 - q(2) = 7;
%e A249477 a(3) = exp(4)*11 - q(3) = 47;
%e A249477 a(4) = exp(4)*56 - q(4) = 360;
%e A249477 a(5) = exp(4)*324 - q(5) = 2884;
%e A249477 a(6) = exp(4)*2324 - q(6) = 26068;
%e A249477 a(7) = exp(4)*18332 - q(7) = 250140;
%e A249477 a(8) = exp(4)*167544 - q(8) = 2659544; ...
%o A249477 (PARI) \p100 \\ set precision
%o A249477 {P=Vec(serlaplace(prod(k=1, 31, 1/(1-x^k/k +O(x^31))))); } \\ A007841
%o A249477 {Q=Vec(serlaplace(sum(n=1, 201, 4^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
%o A249477 for(n=0, 30, print1(round(exp(4)*P[n+1]-Q[n+1]), ", "))
%Y A249477 Cf. A007841, A249078, A249474, A249475, A249476.
%K A249477 nonn
%O A249477 0,3
%A A249477 _Paul D. Hanna_, Oct 29 2014