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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). 3
 1, 1, 6, 64, 1192, 32360, 1257880, 64644520, 4315649600, 360332919360, 36979925855040, 4564758983929920, 668857835862650880, 114624254940995404800, 22742780483191398589440, 5169745984444274224143360, 1335478685859609449305006080, 388956774210908224056394014720 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..100 EXAMPLE 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 +... such that A(x) = exp(1)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n^2) = Sum_{n>=0} A249588(n)*x^n/n!^2, and Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k^2 - x^k). More explicitly, P(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...); 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)) +... We can illustrate the initial terms a(n) in the following manner. The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin: q(0) = 1.718281828459045235360287471352662... q(1) = 1.718281828459045235360287471352662... q(2) = 7.591409142295226176801437356763312... q(3) = 69.19580959449321653265408609628046... q(4) = 1134.849245160942721468406075477879... q(5) = 28464.27419359959618642179245898717... q(6) = 1032370.298622570136419515164963586... q(7) = 50636398.83839730972810740431058131... q(8) = 3247132530.854165002836403983556004... q(9) = 263126229989.7260044371780752021631... and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n^2) begin: A007841 = [1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, ...]; from which we can generate this sequence like so: a(0) = exp(1)*1 - q(0) = 1; a(1) = exp(1)*1 - q(1) = 1; a(2) = exp(1)*5 - q(2) = 6; a(3) = exp(1)*49 - q(3) = 64; a(4) = exp(1)*856 - q(4) = 1192; a(5) = exp(1)*22376 - q(5) = 32360; a(6) = exp(1)*842536 - q(6) = 1257880; a(7) = exp(1)*42409480 - q(7) = 64644520; a(8) = exp(1)*2782192064 - q(8) = 4315649600; ... PROG (PARI) \p100 \\ set precision {P=Vec(serlaplace(serlaplace(prod(k=1, 31, 1/(1-x^k/k^2 +O(x^31)))))); } \\ A249588 {Q=Vec(serlaplace(serlaplace(sum(n=1, 201, prod(k=1, n, 1./(k-x^k/k +O(x^31))))))); } for(n=0, 30, print1(round(exp(1)*P[n+1]-Q[n+1]), ", ")) CROSSREFS Cf. A249590, A249078, A249588. Sequence in context: A141008 A336114 A258425 * A333983 A087488 A249896 Adjacent sequences:  A249589 A249590 A249591 * A249593 A249594 A249595 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 01 2014 STATUS approved

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Last modified November 29 12:24 EST 2021. Contains 349416 sequences. (Running on oeis4.)