A147304 -id:A147304 - cp's OEIS frontend

A147291 a(n) = Sum_{k=1..n^2-1} binomial(2k,k).

0, 28, 17576, 209295260, 43308802158650, 150315393336149895056, 8610524734277600186228691452, 8068213695203463278728832778415607708, 122985780058082302876789680971972469134558550878, 30386103720799858392019761983012781659021124133753353112778

Offset: 1

N. J. A. Sloane, Apr 25 2009

nonn

D. Callan, Divisibility of a Central Binomial Sum: A11292 and A11307, Amer. Math. Monthly, 116 (2009), 468-470.

Cf. A146977, A066796, A147304.

Table[Sum[Binomial[2*k, k], {k, 1, n^2 - 1}], {n, 1, 10}] (* Vaclav Kotesovec, Jun 07 2019 *)

a(n) = sum(k=1, n^2-1, binomial(2*k,k)); \\ Michel Marcus, Jul 05 2018

a(n) ~ 4^(n^2) / (3*sqrt(Pi)*n). - Vaclav Kotesovec, Jun 07 2019

A147303 Numbers k where records occur in expression Log[A147298(k)]/Log[k] k=2,3,4,...

Original entry on oeis.org

2, 3, 6, 7, 14, 15, 22, 30, 42, 62, 66, 70, 78, 102, 114, 158, 166, 182, 186, 210, 222, 230, 255, 258, 282, 318, 330, 402, 430, 438, 462, 470, 474, 494, 498, 510, 570, 582, 598, 690, 710, 786, 798, 822, 870, 906, 930, 942, 1002, 1038, 1074, 1110, 1122, 1146, 1158

Offset: 1

Artur Jasinski, Nov 06 2008

nonn

Limit k->Infinity Log[A147298(k)]/Log[k] = 2.

Values m for which records occur, see A147301.

Cf. A147298-A147302, A147304-A147307.

logmin = 10^10; logmax = 0; aa = {}; bb = {}; cc = {}; dd = {}; ee = {}; ff = {}; hh = {}; ii = {}; jj = {}; Do[min = 10^100; max = 0; Do[If[GCD[m, n, n - m] == 1, s = m n (n - m); k = FactorInteger[s]; g = 1; Do[g = g k[[p]][[1]], {p, 1, Length[k]}]; If[g > max, max = g; mmax = m]; If[g < min, min = g; mmin = m]], {m, 1, n - 1}]; AppendTo[aa, min]; If[(Log[n]/Log[min]) > logmax, logmax = (Log[n]/Log[min]); AppendTo[ee, {N[logmax], n, mmin, min, mmax, max}]; AppendTo[ff, n]]; If[(Log[n]/Log[min]) < logmin, logmin = (Log[n]/Log[min]); AppendTo[hh, {N[logmin], n, mmin, min, mmax, max}]; AppendTo[ii, n]]; AppendTo[bb, max]; AppendTo[dd, mmin], {n, 2, 1200}]; ii

cp's OEIS Frontend

A147291 a(n) = Sum_{k=1..n^2-1} binomial(2k,k).

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