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A249592 E.g.f.: exp(1)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n^2) and Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k - x^k/k).

%I A249592 #7 Jul 29 2023 21:34:25
%S A249592 1,1,6,64,1192,32360,1257880,64644520,4315649600,360332919360,
%T A249592 36979925855040,4564758983929920,668857835862650880,
%U A249592 114624254940995404800,22742780483191398589440,5169745984444274224143360,1335478685859609449305006080,388956774210908224056394014720
%N A249592 E.g.f.: exp(1)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n^2) and Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k - x^k/k).
%H A249592 Paul D. Hanna, <a href="/A249592/b249592.txt">Table of n, a(n) for n = 0..100</a>
%e A249592 E.g.f.: 1 + x + 6*x^2/2!^2 + 64*x^3/3!^2 + 1192*x^4/4!^2 + 32360*x^5/5!^2 +...
%e A249592 such that A(x) = exp(1)*P(x) - Q(x), where
%e A249592 P(x) = 1/Product_{n>=1} (1 - x^n/n^2) = Sum_{n>=0} A249588(n)*x^n/n!^2, and
%e A249592 Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k - x^k/k).
%e A249592 More explicitly,
%e A249592 P(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...);
%e A249592 Q(x) = 1/(1-x) + 1/((1-x)*(2-x^2/2)) + 1/((1-x)*(2-x^2/2)*(3-x^3/3)) + 1/((1-x)*(2-x^2/2)*(3-x^3/3)*(4-x^4/4)) + 1/((1-x)*(2-x^2/2)*(3-x^3/3)*(4-x^4/4)*(5-x^5/5)) +...
%e A249592 We can illustrate the initial terms a(n) in the following manner.
%e A249592 The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
%e A249592 q(0) = 1.718281828459045235360287471352662...
%e A249592 q(1) = 1.718281828459045235360287471352662...
%e A249592 q(2) = 7.591409142295226176801437356763312...
%e A249592 q(3) = 69.19580959449321653265408609628046...
%e A249592 q(4) = 1134.849245160942721468406075477879...
%e A249592 q(5) = 28464.27419359959618642179245898717...
%e A249592 q(6) = 1032370.298622570136419515164963586...
%e A249592 q(7) = 50636398.83839730972810740431058131...
%e A249592 q(8) = 3247132530.854165002836403983556004...
%e A249592 q(9) = 263126229989.7260044371780752021631...
%e A249592 and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n^2) begin:
%e A249592 A007841 = [1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, ...];
%e A249592 from which we can generate this sequence like so:
%e A249592 a(0) = exp(1)*1 - q(0) = 1;
%e A249592 a(1) = exp(1)*1 - q(1) = 1;
%e A249592 a(2) = exp(1)*5 - q(2) = 6;
%e A249592 a(3) = exp(1)*49 - q(3) = 64;
%e A249592 a(4) = exp(1)*856 - q(4) = 1192;
%e A249592 a(5) = exp(1)*22376 - q(5) = 32360;
%e A249592 a(6) = exp(1)*842536 - q(6) = 1257880;
%e A249592 a(7) = exp(1)*42409480 - q(7) = 64644520;
%e A249592 a(8) = exp(1)*2782192064 - q(8) = 4315649600; ...
%o A249592 (PARI) \p100 \\ set precision
%o A249592 {P=Vec(serlaplace(serlaplace(prod(k=1, 31, 1/(1-x^k/k^2 +O(x^31)))))); } \\ A249588
%o A249592 {Q=Vec(serlaplace(serlaplace(sum(n=1, 201, prod(k=1, n, 1./(k-x^k/k +O(x^31))))))); }
%o A249592 for(n=0, 30, print1(round(exp(1)*P[n+1]-Q[n+1]), ", "))
%Y A249592 Cf. A249590, A249078, A249588.
%K A249592 nonn
%O A249592 0,3
%A A249592 _Paul D. Hanna_, Nov 01 2014