The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A248658

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A248658

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^3 * x^(2*k).
(history; published version)

#13 by Paul D. Hanna at Wed Oct 15 21:55:52 EDT 2014

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#12 by Paul D. Hanna at Wed Oct 15 21:55:49 EDT 2014

COMMENTS

Limit a(n)/a(n+1) = ( 1 - t = t^3 = 0.3176721961... where t = ((sqrt(93)+9)/18)^(1/3) - ((sqrt(93)-9)/18)^(1/3) )^3 = 0.3176721961...

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#11 by Paul D. Hanna at Wed Oct 15 21:37:31 EDT 2014

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approved

#10 by Paul D. Hanna at Wed Oct 15 21:37:28 EDT 2014

COMMENTS

What is limit Limit a(n+1)/a(n+1) = ( ((sqrt(93)+9)/18)^(1/3)? Specific value: a - ((sqrt(3000193)-9)/a18)^(300001/3) ) = ^3 = 0.147794111603176721961...

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approved

editing

#9 by Paul D. Hanna at Wed Oct 15 21:16:03 EDT 2014

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editing

approved

#8 by Paul D. Hanna at Wed Oct 15 21:15:58 EDT 2014

COMMENTS

What is limit a(n+1)/a(n)? Specific value: a(30001)/a(30000) = 3.14779411160...

LINKS

Paul D. Hanna, <a href="/A248658/b248658.txt">Table of n, a(n) for n = 0..500</a>

FORMULA

G.f.: A(x) = Sum_{n>=0} (3n3*n)!/(n!)^3 * x^(4*n) / (1-x-x^3)^(3*n+1).

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approved

editing

#7 by Vaclav Kotesovec at Wed Oct 15 16:53:06 EDT 2014

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approved

#6 by Vaclav Kotesovec at Wed Oct 15 16:50:33 EDT 2014

MATHEMATICA

Table[Sum[Binomial[n-2*k, k]^3, {k, 0, Floor[n/3]}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 15 2014 *)

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#5 by Paul D. Hanna at Fri Oct 10 18:04:59 EDT 2014

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approved

#4 by Paul D. Hanna at Fri Oct 10 18:04:57 EDT 2014

NAME

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^3 * x^(2*k).

FORMULA

G.f.: A(x) = Sum_{n>=0} (3n)!/(n!)^3 * x^(4*n) / (1-x-x^3)^(3n3*n+1).

EXAMPLE

G.f. A(x) = 1 + x + x^2 + 2*x^3 + 9*x^4 + 28*x^5 + 66*x^6 + 153*x^7 +...

PROG

(PARI) {a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^3*x^(2*k)) +x*O(x^n)); polcoeff(A, n)}

CROSSREFS

Cf. A181545, A181543.

STATUS

approved

editing