A188675 -id:A188675 - cp's OEIS frontend

A188687 Partial binomial sums of binomial(3n,n)/(2n+1) = A001764(n).

1, 2, 6, 25, 126, 704, 4183, 25897, 165166, 1077520, 7156352, 48222354, 328859011, 2265428728, 15740837575, 110187356134, 776336572878, 5501042194580, 39177463572112, 280277949384146, 2013277273220064, 14514764553512488, 104993261648226446

Offset: 0

Emanuele Munarini, Apr 08 2011

Nathaniel Johnston, Table of n, a(n) for n = 0..400
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Cf. A005809, A001764, A188675, A188676, A104859, A188678, A188679, A188680, A188681, A188682, A188683, A188684, A188685, A188686.

Table[Sum[Binomial[n,k]Binomial[3k,k]/(2k+1),{k,0,n}],{n,0,22}]

makelist(sum(binomial(n,k)*binomial(3*k,k)/(2*k+1),k,0,n),n,0,20);

a(n) = Sum_{k=0..n} binomial(n,k)*binomial(3k,k)/(2k+1).
G.f.: (2/sqrt(3x*(1-x)))*sin((1/3)*arcsin(3/2*sqrt(3*x/(1-x)))).
Recurrence: 2*n*(2*n+1)*a(n) = (39*n^2-35*n+8)*a(n-1) - 2*(n-1)*(33*n-32)*a(n-2) + 31*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 31^(n+3/2)/(3^4*2^(2*n+2)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012
G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x) * A(x)^3. - Ilya Gutkovskiy, Jul 25 2021

A263134 a(n) = Sum_{k=0..n} binomial(3*k+1,k).

Original entry on oeis.org

1, 5, 26, 146, 861, 5229, 32361, 202905, 1284480, 8191380, 52543545, 338641305, 2191124301, 14224347181, 92603307541, 604342068085, 3952451061076, 25898039418496, 169977746765071, 1117287239602471, 7353933943361866, 48461930821297546

Offset: 0

Bruno Berselli, Oct 10 2015

Primes in sequence: 5, 92603307541, 52176309488123582020412161, ...

a(n) is divisible by n for n = 1, 2, 8, 55, 82, 171, 210, 1060, 1141, ...

Bruno Berselli and G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 form Bruno Berselli)

Partial sums of A045721.
Cf. A079309: Sum_{k=0..n} binomial(2*k+1,k).
Cf. A188675: Sum_{k=0..n} binomial(3*k,k).
Cf. A087413: Sum_{k=0..n} binomial(3*k+2,k).

[&+[Binomial(3*k+1,k): k in [0..n]]: n in [0..25]];

Table[Sum[Binomial[3 k + 1, k], {k, 0, n}], {n, 0, 25}]

makelist(sum(binomial(3*k+1,k),k,0,n),n,0,25);

a(n) = sum(k=0, n, binomial(3*k+1,k)) \\ Colin Barker, Oct 16 2015

[sum(binomial(3*k+1,k) for k in (0..n)) for n in (0..25)]

Recurrence: 2*n*(2*n + 1)*a(n) = (31*n^2 + 2*n - 3)*a(n-1) - 3*(3*n - 1)*(3*n + 1)*a(n-2). - Vaclav Kotesovec, Oct 11 2015

a(n) ~ 27^(n + 3/2)/(23*sqrt(Pi*n)*4^(n + 1)). - Vaclav Kotesovec, Oct 11 2015

A087413 a(n) = Sum_{k=1..n} C(3*k,k)/3.

Original entry on oeis.org

1, 6, 34, 199, 1200, 7388, 46148, 291305, 1853580, 11868585, 76380825, 493606725, 3201081873, 20821158233, 135776966761, 887393271310, 5811082966885, 38119865826420, 250447855600320, 1647729357535485, 10854207824989830, 71581930485576630

Offset: 1

Benoit Cloitre, Oct 21 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A188675: Sum_{k=0..n} binomial(3*k,k).

Cf. A263134: Sum_{k=0..n} binomial(3*k+1,k).

Table[Sum[Binomial[3*k,k]/3,{k,1,n}],{n,1,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

a(n)=sum(k=1,n,binomial(3*k,k))/3 \\ Charles R Greathouse IV, Nov 10 2011

a=vector(99,i,1);for(n=2,#a,a[n]=a[n-1]+binomial(3*n,n)/3);a \\ Charles R Greathouse IV, Nov 10 2011

G.f.: 1/((3*g-1)*(g^3-2*g^2+g-1)*(g-1)^2) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011
Recurrence: 2*n*(2*n-1)*a(n) = (31*n^2-29*n+6)*a(n-1) - 3*(3*n-2)*(3*n-1)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 3^(3*n+5/2)/(23*2^(2*n+1)*sqrt(Pi)*sqrt(n)). - Vaclav Kotesovec, Oct 14 2012
a(n) = Sum_{k=1..n} binomial(3*k-1,k-1). [Bruno Berselli, Oct 10 2015]

A225612 Partial sums of the binomial coefficients C(4*n,n).

Original entry on oeis.org

1, 5, 33, 253, 2073, 17577, 152173, 1336213, 11854513, 105997793, 953658321, 8622997453, 78291531921, 713305091521, 6518037055321, 59712126248041, 548239063327621, 5043390644753269, 46475480410336709, 428936432074181109, 3964252574286355429

Offset: 0

Vaclav Kotesovec, Aug 06 2013

Generally (for p>1), partial sums of the binomial coefficients C(p*n,n) are asymptotic to (1/(1-(p-1)^(p-1)/p^p)) * sqrt(p/(2*Pi*n*(p-1))) * (p^p/(p-1)^(p-1))^n.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A006134 (p=2), A188675 (p=3), A225615 (p=5).

A225612:=n->add(binomial(4*k,k), k=0..n): seq(A225612(n), n=0..30); # Wesley Ivan Hurt, Apr 01 2017

Table[Sum[Binomial[4*k, k], {k, 0, n}], {n, 0, 20}]
Accumulate[Table[Binomial[4n,n],{n,0,20}]] (* Harvey P. Dale, Feb 01 2015 *)

for(n=0,50, print1(sum(k=0,n, binomial(4*k,k)), ", ")) \\ G. C. Greubel, Apr 01 2017

Recurrence: 3*n*(3*n-2)*(3*n-1)*a(n) = (283*n^3 - 411*n^2 + 182*n - 24)*a(n-1) - 8*(2*n-1)*(4*n-3)*(4*n-1)*a(n-2).

a(n) ~ 2^(8*n+17/2)/(229*sqrt(Pi*n)*3^(3*n+1/2)).

A225615 Partial sums of the binomial coefficients C(5*n,n).

Original entry on oeis.org

1, 6, 51, 506, 5351, 58481, 652256, 7376776, 84281461, 970444596, 11242722766, 130896288616, 1530255133591, 17951328648871, 211205085558031, 2491217772274111, 29449438902782636, 348806466779875961, 4138454609488474736, 49176494325141603881

Offset: 0

Vaclav Kotesovec, Aug 06 2013

Generally (for p>1), partial sums of the binomial coefficients C(p*n,n) are asymptotic to (1/(1-(p-1)^(p-1)/p^p)) * sqrt(p/(2*Pi*n*(p-1))) * (p^p/(p-1)^(p-1))^n.

G. C. Greubel, Table of n, a(n) for n = 0..900

Cf. A006134 (p=2), A188675 (p=3), A225612 (p=4).

A225615:=n->add(binomial(5*k,k), k=0..n): seq(A225615(n), n=0..30); # Wesley Ivan Hurt, Apr 01 2017

Table[Sum[Binomial[5*k, k], {k, 0, n}], {n, 0, 20}]

for(n=0,50, print1(sum(k=0,n, binomial(5*k,k)), ", ")) \\ G. C. Greubel, Apr 01 2017

Recurrence: 8*n*(2*n-1)*(4*n-3)*(4*n-1)*a(n) = (3381*n^4 - 6634*n^3 + 4551*n^2 - 1274*n + 120)*a(n-1) - 5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-2).

a(n) ~ 5^(5*n+11/2)/(2869*sqrt(Pi*n)*2^(8*n+3/2)).

A356282 a(n) = Sum_{k=0..n} binomial(3n, n-k) p(k), where p(k) is the partition function A000041.

Original entry on oeis.org

1, 4, 23, 141, 888, 5675, 36602, 237563, 1548995, 10135554, 66504699, 437359454, 2881641263, 19016505326, 125664684700, 831400186740, 5506287269802, 36501297800013, 242167539749593, 1607851773270316, 10682384379036741, 71016046921543562, 472376627798814453

Offset: 0

Vaclav Kotesovec, Aug 01 2022

nonn

Cf. A000041, A188675, A356280, A356283.

Table[Sum[PartitionsP[k]*Binomial[3*n, n-k], {k, 0, n}], {n, 0, 30}]

a(n) = sum(k=0, n, binomial(3*n, n-k)*numbpart(k)); \\ Michel Marcus, Aug 02 2022

a(n) ~ c * 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n + 1)), where c = Sum_{j>=0} p(j)/2^j = A065446 = 3.4627466194550636115379573429...

A356283 a(n) = Sum_{k=0..n} binomial(3n, n-k) q(k), where q(k) is the number of partitions into distinct parts (A000009).

Original entry on oeis.org

1, 4, 22, 131, 807, 5066, 32188, 206242, 1329733, 8614685, 56024538, 365491218, 2390613557, 15671221522, 102925324569, 677110860689, 4460956827127, 29427611146335, 194348311824025, 1284856925961827, 8502252246841668, 56309476194587377, 373220349572126265

Offset: 0

Vaclav Kotesovec, Aug 01 2022

nonn

Cf. A000009, A188675, A356281, A356282.

Table[Sum[PartitionsQ[k]*Binomial[3*n, n-k], {k, 0, n}], {n, 0, 30}]

a(n) ~ c * 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n + 1)), where c = Sum_{j>=0} q(j)/2^j = A079555 = 2.384231029031371724149899288678...

A358146 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(k*j,j).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 9, 4, 1, 1, 5, 19, 29, 5, 1, 1, 6, 33, 103, 99, 6, 1, 1, 7, 51, 253, 598, 351, 7, 1, 1, 8, 73, 506, 2073, 3601, 1275, 8, 1, 1, 9, 99, 889, 5351, 17577, 22165, 4707, 9, 1, 1, 10, 129, 1429, 11515, 58481, 152173, 138445, 17577, 10, 1

Offset: 0

Seiichi Manyama, Oct 31 2022

			Square array begins:
  1, 1,   1,    1,     1,     1, ...
  1, 2,   3,    4,     5,     6, ...
  1, 3,   9,   19,    33,    51, ...
  1, 4,  29,  103,   253,   506, ...
  1, 5,  99,  598,  2073,  5351, ...
  1, 6, 351, 3601, 17577, 58481, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give: A000012, A001477(n+1), A006134, A188675, A225612, A225615.
Main diagonal gives A226391.
Cf. A358050.

T(n, k) = sum(j=0, n, binomial(k*j, j));

A359665 a(n) = Sum_{k=0..n} binomial(k^3, k).

Original entry on oeis.org

1, 2, 30, 2955, 638331, 235169606, 131748994154, 104332124742623, 110963563379491743, 152605484049946645638, 263562165946020159478038, 558488792578762177358255808, 1424733420462958066911824023728, 4307309064570490624823548890385698, 15228800547242034570505949850130312826

Offset: 0

Vaclav Kotesovec, Jan 10 2023

nonn

Cf. A005809, A107444, A188675, A295773.

Table[Sum[Binomial[k^3, k], {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(k^3, k)); \\ Michel Marcus, Jan 10 2023

a(n) ~ exp(n) * n^(2*n - 1/2) / sqrt(2*Pi).

A356284 a(n) = Sum_{k=0..n} binomial(3n, k) p(k), where p(k) is the partition function A000041.

Original entry on oeis.org

1, 4, 37, 334, 3280, 29437, 282253, 2517904, 23209785, 206685325, 1858085653, 16266231810, 144339750406, 1250038867329, 10882952174845, 93546973843450, 804847296088574, 6843680884286307, 58300294406199829, 491683063753997014, 4148296662116385627, 34746182976196757434

Offset: 0

Vaclav Kotesovec, Aug 01 2022

nonn

Cf. A000041, A188675, A356267, A356285.

Table[Sum[Binomial[3*n, k] * PartitionsP[k], {k, 0, n}], {n, 0, 30}]

a(n) = sum(k=0, n, binomial(3*n, k)*numbpart(k)); \\ Michel Marcus, Aug 02 2022

a(n) ~ 3^(3*n) * exp(Pi*sqrt(2*n/3)) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 2)).

cp's OEIS Frontend

A188687 Partial binomial sums of binomial(3n,n)/(2n+1) = A001764(n).

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A263134 a(n) = Sum_{k=0..n} binomial(3*k+1,k).

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A087413 a(n) = Sum_{k=1..n} C(3*k,k)/3.

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A225612 Partial sums of the binomial coefficients C(4*n,n).

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A225615 Partial sums of the binomial coefficients C(5*n,n).

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A356282 a(n) = Sum_{k=0..n} binomial(3*n, n-k) * p(k), where p(k) is the partition function A000041.

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A356283 a(n) = Sum_{k=0..n} binomial(3*n, n-k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).

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A358146 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(k*j,j).

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A356282 a(n) = Sum_{k=0..n} binomial(3n, n-k) p(k), where p(k) is the partition function A000041.

A356283 a(n) = Sum_{k=0..n} binomial(3n, n-k) q(k), where q(k) is the number of partitions into distinct parts (A000009).

A356284 a(n) = Sum_{k=0..n} binomial(3n, k) p(k), where p(k) is the partition function A000041.