The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A303056 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A303056

G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - A(x))^n.
(history; published version)

#38 by Paul D. Hanna at Fri Sep 29 10:30:15 EDT 2023

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#37 by Paul D. Hanna at Fri Sep 29 10:30:13 EDT 2023

CROSSREFS

Cf. A304642, A304639, A303926.

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approved

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#36 by Vaclav Kotesovec at Sat Sep 26 08:08:43 EDT 2020

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#35 by Vaclav Kotesovec at Sat Sep 26 08:08:33 EDT 2020

FORMULA

a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.1610886538654... and c = 0.11739505492506... - Vaclav Kotesovec, Sep 26 2020

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approved

editing

#34 by Paul D. Hanna at Sun Sep 20 10:04:09 EDT 2020

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approved

#33 by Paul D. Hanna at Sun Sep 20 10:03:54 EDT 2020

CROSSREFS

Cf. A337755, A337756, A337757.

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approved

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#32 by Paul D. Hanna at Sat Jun 22 22:25:03 EDT 2019

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#31 by Paul D. Hanna at Sat Jun 22 22:24:59 EDT 2019

COMMENTS

More generally, the following sums are equal:

(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,

(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),

for any fixed integer k; here, k = 1 with r = 1, p = -A(x), q = (1+x). - Paul D. Hanna, Jun 22 2019

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approved

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#30 by Paul D. Hanna at Sat Jun 22 22:19:15 EDT 2019

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approved

#29 by Paul D. Hanna at Sat Jun 22 22:19:13 EDT 2019

CROSSREFS

Cf. A326282, A326283, A326284.

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approved

editing