This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A249475 #7 Oct 29 2014 22:19:37
%S A249475 1,1,5,25,156,1048,8400,72384,710184,7519240,87797880,1098513880,
%T A249475 14945280640,216079283040,3352657547680,55071779464352,
%U A249475 961293645943680,17669716422651776,342988501737128576,6978772157389361280,149123855108936024576,3328674238745847019520
%N A249475 E.g.f.: exp(2)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} 2^n/Product_{k=1..n} (k - x^k).
%C A249475 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.
%H A249475 Paul D. Hanna, <a href="/A249475/b249475.txt">Table of n, a(n) for n = 0..100</a>
%e A249475 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 156*x^4/4! + 1048*x^5/5! +...
%e A249475 such that A(x) = exp(2)*P(x) - Q(x), where
%e A249475 P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
%e A249475 Q(x) = Sum_{n>=1} 2^n / Product_{k=1..n} (k - x^k).
%e A249475 More explicitly,
%e A249475 P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...);
%e A249475 Q(x) = 2/(1-x) + 2^2/((1-x)*(2-x^2)) + 2^3/((1-x)*(2-x^2)*(3-x^3)) + 2^4/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 2^5/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +...
%e A249475 We can illustrate the initial terms a(n) in the following manner.
%e A249475 The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
%e A249475 q(0) = 6.3890560989306502272...
%e A249475 q(1) = 6.3890560989306502272...
%e A249475 q(2) = 17.167168296791950681...
%e A249475 q(3) = 56.279617088237152499...
%e A249475 q(4) = 257.78714154011641272...
%e A249475 q(5) = 1346.0541760535306736...
%e A249475 q(6) = 8772.1663739148311280...
%e A249475 q(7) = 63072.176405596679965...
%e A249475 q(8) = 527808.01503923686167...
%e A249475 q(9) = 4851990.6204200261720...
%e A249475 and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n) begin:
%e A249475 A007841 = [1, 1, 3, 11, 56, 324, 2324, 18332, 167544, ...];
%e A249475 from which we can generate this sequence like so:
%e A249475 a(0) = exp(2)*1 - q(0) = 1;
%e A249475 a(1) = exp(2)*1 - q(1) = 1;
%e A249475 a(2) = exp(2)*3 - q(2) = 5;
%e A249475 a(3) = exp(2)*11 - q(3) = 25;
%e A249475 a(4) = exp(2)*56 - q(4) = 156;
%e A249475 a(5) = exp(2)*324 - q(5) = 1048;
%e A249475 a(6) = exp(2)*2324 - q(6) = 8400;
%e A249475 a(7) = exp(2)*18332 - q(7) = 72384;
%e A249475 a(8) = exp(2)*167544 - q(8) = 710184; ...
%o A249475 (PARI) \p100 \\ set precision
%o A249475 {P=Vec(serlaplace(prod(k=1, 31, 1/(1-x^k/k +O(x^31))))); } \\ A007841
%o A249475 {Q=Vec(serlaplace(sum(n=1, 201, 2^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
%o A249475 for(n=0, 30, print1(round(exp(2)*P[n+1]-Q[n+1]), ", "))
%Y A249475 Cf. A007841, A249078, A249474, A249476, A249477, A249478, A249480.
%K A249475 nonn
%O A249475 0,3
%A A249475 _Paul D. Hanna_, Oct 29 2014