The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A349293 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A349293

G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(x)^7)).
(history; published version)

#15 by Vaclav Kotesovec at Sun Nov 14 05:06:43 EST 2021

STATUS

editing

approved

#14 by Vaclav Kotesovec at Sun Nov 14 05:06:39 EST 2021

COMMENTS

In general, for k>=1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial((k+1)*j,j) / (k*j+1) ~ sqrt(1 + (k-1)*r) / ((k+1)^(1/k) * sqrt(2*k*(k+1)*Pi*(1-r)) * n^(3/2) * r^(n + 1/k)), where r is the smallest real root of the equation (k+1)^(k+1) * r = k^k * (1-r)^k. - Vaclav Kotesovec, Nov 14 2021

STATUS

approved

editing

#13 by Vaclav Kotesovec at Sun Nov 14 05:03:01 EST 2021

STATUS

editing

approved

#12 by Vaclav Kotesovec at Sun Nov 14 04:57:48 EST 2021

COMMENTS

In general, for k>=1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial((k+1)*j,j) / (k*j+1) ~ sqrt(1 + (k-1)*r) / ((k+1)^(1/k) * sqrt(2*k*(k+1)*Pi*(1-r)) * n^(3/2) * r^(n + 1/k)), where r is the root of the equation (k+1)^(k+1) * r = k^k * (1-r)^k. - Vaclav Kotesovec, Nov 14 2021

FORMULA

a(n) ~ sqrt(1 + 6*r) / (2^(17/7) * sqrt(7*Pi*(1-r)) * n^(3/2) * r^(n + 1/7)), where r = 0.0375502499742240443056934699070050852345109331376051496159609551... is the real root of the equation 8^8 * r = 7^7 * (1-r)^7. - Vaclav Kotesovec, Nov 14 2021

STATUS

approved

editing

#11 by N. J. A. Sloane at Sun Nov 14 01:20:55 EST 2021

STATUS

reviewed

approved

#10 by Joerg Arndt at Sun Nov 14 01:11:56 EST 2021

STATUS

proposed

reviewed

#9 by Michel Marcus at Sun Nov 14 00:28:14 EST 2021

STATUS

editing

proposed

#8 by Michel Marcus at Sun Nov 14 00:28:12 EST 2021

PROG

(PARI) a(n) = sum(k=0, n, binomial(n+6*k, 7*k) * binomial(8*k, k) / (7*k+1)); \\ Michel Marcus, Nov 14 2021

STATUS

proposed

editing

#7 by Seiichi Manyama at Sat Nov 13 22:03:19 EST 2021

STATUS

editing

proposed

#6 by Seiichi Manyama at Sat Nov 13 21:50:16 EST 2021

LINKS

Seiichi Manyama, <a href="/A349293/b349293.txt">Table of n, a(n) for n = 0..500</a>

STATUS

approved

editing