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#20 by Vaclav Kotesovec at Sat Dec 23 03:43:52 EST 2017
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#19 by Vaclav Kotesovec at Sat Dec 23 03:41:28 EST 2017
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| FORMULA
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a(n) ~ c * d^n * n^(2*n-2) / exp(2*n), where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = -0.1875440087... - Vaclav Kotesovec, Dec 23 2017
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| STATUS
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approved
editing
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#18 by Paul D. Hanna at Sat Dec 16 18:32:13 EST 2017
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#17 by Paul D. Hanna at Sat Dec 16 18:32:10 EST 2017
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#16 by Paul D. Hanna at Wed Dec 13 22:10:21 EST 2017
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#15 by Paul D. Hanna at Wed Dec 13 22:10:17 EST 2017
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| COMMENTS
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Surprisingly, the logarithm of the e.g.f. is an integer series (A296171).
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| STATUS
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approved
editing
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#14 by Paul D. Hanna at Wed Dec 13 22:07:55 EST 2017
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#13 by Paul D. Hanna at Wed Dec 13 22:07:47 EST 2017
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| COMMENTS
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Surprisingly, the logarithm of the e.g.f. is an integer series (A296171).
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| EXAMPLE
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[1, 1, 7, 154, 7609, 695856, 103805719, 23134327168, 7227250033329, 3017857024161280, 1623903877812828871, ...]., ..., A296232(n), ...].
log(A(x)) = x - x^2 - x^3 - 9*x^4 - 134*x^5 - 2852*x^6 - 79096*x^7 - 2699480*x^8 - 109201844*x^9 - 5100872244*x^10 - 269903909820*x^11 - 15944040740604*x^12 - 1039553309158964*x^13 - 74123498185170292*x^14 - 5736368141560365292*x^15 - 478780244956262592748*x^16 - 42865943103053965559668*x^17 - 4097785410628237071311764*x^18 - 416572537937169684523985420*x^19 - 44873737158384968851319470220*x^20 +...+ A296171(n)*x^n +...
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| CROSSREFS
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Cf. A296171 (log), A296232, A296172, A296174, A296176.
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| STATUS
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approved
editing
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#12 by Paul D. Hanna at Fri Dec 08 21:09:13 EST 2017
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#11 by Paul D. Hanna at Fri Dec 08 21:09:10 EST 2017
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| FORMULA
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_ log(A(x)) = Sum_{n>=1} A296171(n) * x^n.
E.g.f. A(x) satisfies:
_ 1/n! * d^n/dx^n A(x)^(n^2) = 1/(n-1)! * d^(n-1)/dx^(n-1) A(x)^(n^2) for n>=1, when evaluated at x = 0.
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| STATUS
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approved
editing
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