The On-Line Encyclopedia of Integer Sequences (OEIS)

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A279013

a(n) = Sum_{k=0..n} binomial(2*k,k)/(k+1)*binomial(2*n-1,n-k).
(history; published version)

#12 by Vaclav Kotesovec at Fri Nov 19 08:09:17 EST 2021

STATUS

editing

approved

#11 by Vaclav Kotesovec at Fri Nov 19 08:09:11 EST 2021

FORMULA

a(n) ~ 5^(2*n + 1/2) / (3^(3/2) * sqrt(Pi) * n^(3/2) * 2^(2*n - 2)). - Vaclav Kotesovec, Nov 19 2021

STATUS

approved

editing

#10 by N. J. A. Sloane at Thu Jan 30 21:29:17 EST 2020

FORMULA

Conjecture D-finite with recurrence: 2*n*(n+1)*(2*n-3)*a(n) -n*(101*n^2-312*n+203)*a(n-1) +(995*n^3-5570*n^2+9567*n-4984)*a(n-2) +2*(-2393*n^3+19300*n^2-50494*n+42835)*a(n-3) +4*(2*n7)*(1408*n^2-9889*n+16690)*a(n-4) -2600*(n-5)*(2*n-7)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Jan 27 2020

Discussion

Thu Jan 30

21:29

OEIS Server: https://oeis.org/edit/global/2847

#9 by R. J. Mathar at Mon Jan 27 13:30:18 EST 2020

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editing

approved

#8 by R. J. Mathar at Mon Jan 27 13:30:10 EST 2020

FORMULA

G.f.: (2*(1-sqrt(1-((1-sqrt(1-4*x))^2)/x))*(1/sqrt(1-4*x)+1)/2*x)/((1-sqrt(1-4*x))^2).

Conjecture D-finite: 2*n*(n+1)*(2*n-3)*a(n) -n*(101*n^2-312*n+203)*a(n-1) +(995*n^3-5570*n^2+9567*n-4984)*a(n-2) +2*(-2393*n^3+19300*n^2-50494*n+42835)*a(n-3) +4*(2*n7)*(1408*n^2-9889*n+16690)*a(n-4) -2600*(n-5)*(2*n-7)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Jan 27 2020

STATUS

approved

editing

#7 by Harvey P. Dale at Wed Feb 06 14:02:35 EST 2019

STATUS

editing

approved

#6 by Harvey P. Dale at Wed Feb 06 14:02:32 EST 2019

MATHEMATICA

Table[Sum[Binomial[2k, k]/(k+1) Binomial[2n-1, n-k], {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Feb 06 2019 *)

STATUS

approved

editing

#5 by N. J. A. Sloane at Sat Dec 03 12:16:10 EST 2016

STATUS

proposed

approved

#4 by Vladimir Kruchinin at Sat Dec 03 06:43:31 EST 2016

STATUS

editing

proposed

#3 by Vladimir Kruchinin at Sat Dec 03 06:43:17 EST 2016

DATA

1, 2, 8, 35, 161, 768, 3773, 19006, 97840, 513264, 2737121, 14805805, 81082383, 448805300, 2507310567, 14120503129, 80082573017, 456977964520, 2621830478785, 15114658956625, 87508451311125, 508589225952740, 2966098696204660, 17352591951620625, 101807952163378141, 598869092459293628, 3531201823416200663, 20867594755738967081