A096131 -id:A096131 - cp's OEIS frontend

A226391 a(n) = Sum_{k=0..n} binomial(k*n, k).

1, 2, 9, 103, 2073, 58481, 2101813, 91492906, 4671050401, 273437232283, 18046800575211, 1325445408799007, 107200425419863009, 9466283137384124247, 906151826270369213655, 93459630239922214535911, 10331984296666203358431361, 1218745075041575200343722415

Offset: 0

Vaclav Kotesovec, Jun 06 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A072034, A086331, A096131, A167008, A167010, A349471.

[(&+[Binomial(n*j,j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Aug 31 2022

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

A226391(n):=sum(binomial(k*n,k), k,0,n); makelist(A226391(n),n,0,30); /* Martin Ettl, Jun 06 2013 */

@CachedFunction
def A226391(n): return sum(binomial(n*j, j) for j in (0..n))
[A226391(n) for n in (0..30)] # G. C. Greubel, Aug 31 2022

a(n) ~ binomial(n^2, n).

A361281 a(n) = n! * Sum_{k=0..n} binomial(n*k,n-k)/k!.

Original entry on oeis.org

1, 1, 5, 37, 481, 10001, 288901, 10820965, 511186817, 29843419681, 2106779832901, 176180844038981, 17165338119936865, 1924030148121500017, 245630480526435293381, 35409038825312233143301, 5719025066628373334423041, 1027649751647068260334391105

Offset: 0

Seiichi Manyama, Mar 06 2023

nonn

From  Peter Bala, Mar 12 2023: (Start)
It appears that a(n) == 1 (mod 4) and a(5*n+2) == 0 (mod 5) for all n. More generally we conjecture that a(n+k) == a(n) (mod k) for all n and k. If true, then for each k, the sequence a(n) taken modulo k is a periodic sequence and the period divides k.
Let F(x) and G(x) be power series with integer coefficients with G(0) = 1. Define b(n) = n! * [x^n] F(x)*exp(x*G(x)^n). Then we conjecture that b(n+k) == b(n) (mod k) for all n and k. The present sequence is the case F(x) = 1, G(x) = 1 + x. Cf. A278070. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..275

Main diagonal of A361277.

Cf. A096131, A099237, A226391, A278070, A293013.

a(n) = n!*sum(k=0, n, binomial(n*k, n-k)/k!);

a(n) = n! * [x^n] exp(x * (1+x)^n).

log(a(n)) ~ n*(2*log(n) - log(log(n)) - 1 - log(2) + log(log(n))/log(n) + 1/(2*log(n)) + log(2)/log(n) - 1/(8*log(n)^2)). - Vaclav Kotesovec, Mar 12 2023

A096130 Triangle read by rows: T(n,k) = binomial(k*n,n), 1 <= k <= n.

Original entry on oeis.org

1, 1, 6, 1, 20, 84, 1, 70, 495, 1820, 1, 252, 3003, 15504, 53130, 1, 924, 18564, 134596, 593775, 1947792, 1, 3432, 116280, 1184040, 6724520, 26978328, 85900584, 1, 12870, 735471, 10518300, 76904685, 377348994, 1420494075, 4426165368, 1, 48620, 4686825, 94143280, 886163135, 5317936260, 23667689815, 85113005120, 260887834350

Offset: 1

Amarnath Murthy, Jul 04 2004

			Triangle begins:
  1;
  1,   6;
  1,  20,   84;
  1,  70,  495,  1820;
  1, 252, 3003, 15504, 53130;
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Row-sums give A096131. The leading diagonal is A014062. Cf. A096131.

Cf. A007318.

Flat(List([1..10],n->List([1..n],k->Binomial(k*n,n)))); # Muniru A Asiru, Aug 12 2018

a:=(n,k)->binomial(k*n,n): seq(seq(a(n,k),k=1..n),n=1..10); # Muniru A Asiru, Aug 12 2018

tabl(nrows) = {for (n=1, nrows, for (k=1, n, print1(binomial(k*n, n), ", ");); print(););} \\ Michel Marcus, May 14 2013

T(n, 1) = 1;
T(n, 2) = A000984(n) for n > 1;
T(n, 3) = A005809(n) for n > 2;
T(n, 4) = A005810(n) for n > 3;
T(n, n) = A014062(n).

Corrected and extended by Reinhard Zumkeller, Jan 09 2005

A336513 a(n) = Sum_{i=1..n} Product_{j=(i-1)n+1..in} j.

Original entry on oeis.org

1, 14, 630, 57264, 8626800, 1940869440, 609372212400, 254507088084480, 136432540607489280, 91303132805512992000, 74605774050426543724800, 73097485140038735163110400, 84588938888439008795675904000, 114144070365165930162064530739200, 177648727691418999798103189989120000

Offset: 1

Seiichi Manyama, Jul 23 2020

			a(2) = 1*2 + 3*4 = 14.
a(3) = 1*2*3 + 4*5*6 + 7*8*9 = 630.
a(4) = 1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16 = 57264.

Seiichi Manyama, Table of n, a(n) for n = 1..214

Main diagonal of A333446.

Cf. A096131, A336502.

a[n_] := n! * Sum[Binomial[k*n, n], {k, 1, n}]; Array[a, 15] (* Amiram Eldar, May 01 2021 *)

{a(n) = sum(i=1, n, prod(j=(i-1)*n+1,i*n, j))}

a(n) = n! * A096131(n) = n! * Sum_{k=1..n} binomial(k*n, n).

A349470 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(k*n,n).

Original entry on oeis.org

1, 1, 5, 65, 1394, 40378, 1470972, 64575585, 3315911300, 194921240846, 12905391110105, 950172113032181, 77000666619646717, 6810514097879311450, 652810277600420281734, 67407087759052608218945, 7459157975936646185855880, 880616251774021869817185430

Offset: 0

Seiichi Manyama, Nov 19 2021

nonn

			a(1) = (1/1!) * (1) = 1.
a(2) = (1/2!) * (-1*2 + 3*4) = 5.
a(3) = (1/3!) * (1*2*3 - 4*5*6 + 7*8*9) = 65.
a(4) = (1/4!) * (-1*2*3*4 + 5*6*7*8 - 9*10*11*12 + 13*14*15*16) = 1394.

Seiichi Manyama, Table of n, a(n) for n = 0..338

Cf. A096130, A096131, A349471, A349480.

a[n_] := Sum[(-1)^(n - k) * Binomial[k*n, n], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2021 *)

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(k*n, n));

a(n) = sum(j=0, n, (-1)^(n-j)*prod(k=(j-1)*n+1, j*n, k))/n!;

a(n) = (1/n!) * Sum_{j=0..n} (-1)^(n-j) * Product_{k=(j-1)*n+1..j*n} k.

a(n) ~ exp(n + 1/2) * n^(n - 1/2) / (sqrt(2*Pi) * (1 + exp(1))). - Vaclav Kotesovec, Nov 20 2021

A029848 a(n) = 1 + C(2n,n) + C(3n,n).

Original entry on oeis.org

3, 6, 22, 105, 566, 3256, 19489, 119713, 748342, 4735446, 30229772, 194242153, 1254381857, 8132826045, 52900345681, 345022543105, 2255449994038, 14773402692946, 96935423713906, 637019314585501, 4191982352334316, 27619973660237476, 182185272080724121, 1202945209263916561

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000984, A096131.

List([0..20],n->1+Binomial(2*n,n)+Binomial(3*n,n)); # Muniru A Asiru, Aug 12 2018

a:=n->1+binomial(2*n,n)+binomial(3*n,n): seq(a(n),n=0..20); # Muniru A Asiru, Aug 12 2018

Table[1+Binomial[2n,n]+Binomial[3n,n],{n,0,30}] (* Harvey P. Dale, Jan 09 2019 *)

a(n) = 1 + binomial(2*n, n) + binomial(3*n, n); \\ Michel Marcus, Aug 12 2018

G.f.: 1/(1-x)+1/sqrt(1-4*x)+cos(arcsin((3^(3/2)*sqrt(x))/2)/3)/sqrt(1-(27*x)/4). - Vladimir Kruchinin, Aug 07 2025

A275779 a(n) = (2^(n^2) - 1)/(1 - 1/2^n).

Original entry on oeis.org

2, 20, 584, 69904, 34636832, 69810262080, 567382630219904, 18519084246547628288, 2422583247133816584929792, 1268889750375080065623288448000, 2659754699919401766201267083003561984, 22306191045953951743035482794815064402563072

Offset: 1

Olivier Gérard, Aug 08 2016

Sum of the geometric progression of ratio 2^n.

Number of all partial binary matrices with rows of length n: A partial binary matrix has 1<=k<=n rows of length n. The number of different partial matrices with k rows is 2^(k*n). a(n) is the sum for k between 1 and n.

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A128889 (accepting the null matrix and excluding the full n*n matrices)

Cf. A096131, A057524.

Table[(2^(n^2) - 1)/(1 - 1/2^n), {n, 1, 10}]

a(n) = {(2^(n^2) - 1)/(1 - 1/2^n)} \\ Andrew Howroyd, Apr 26 2020

a(n) = Sum_{k=1..n} 2^(k*n).

Terms a(11) and beyond from Andrew Howroyd, Apr 26 2020

A323663 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is Sum_{j=1..n} binomial(j*k, k).

Original entry on oeis.org

1, 1, 3, 1, 7, 6, 1, 21, 22, 10, 1, 71, 105, 50, 15, 1, 253, 566, 325, 95, 21, 1, 925, 3256, 2386, 780, 161, 28, 1, 3433, 19489, 18760, 7231, 1596, 252, 36, 1, 12871, 119713, 154085, 71890, 17857, 2926, 372, 45, 1, 48621, 748342, 1303753, 747860, 214396, 38332, 4950, 525, 55

Offset: 1

Seiichi Manyama, Jan 23 2019

			Square array begins:
    1,   1,    1,     1,       1,        1, ...
    3,   7,   21,    71,     253,      925, ...
    6,  22,  105,   566,    3256,    19489, ...
   10,  50,  325,  2386,   18760,   154085, ...
   15,  95,  780,  7231,   71890,   747860, ...
   21, 161, 1596, 17857,  214396,  2695652, ...
   28, 252, 2926, 38332,  539028,  7941438, ...
   36, 372, 4950, 74292, 1197036, 20212950, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns 1-3 give A000217, A002412, A116689.
Rows 1-3 give A000012, A244174, A029848.
Main diagonal is A096131.
Cf. A060539.

A359842 a(n) = Sum_{k=0..n} binomial(n*k,n+k).

Original entry on oeis.org

1, 0, 1, 90, 13690, 3443275, 1308315371, 701623884514, 505274768721332, 470638793249281593, 550707386335951810915, 790898932162231992184327, 1367864138835420575101044139, 2804370191530797723173615407860, 6725366044028696102055907486691290

Offset: 0

Vaclav Kotesovec, Jan 15 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..205

Cf. A096131, A099237, A226391.

a := proc (n) option remember; add(binomial(n*k, n+k), k = 0..n) end:
seq(a(n), n = 0..20); # Peter Bala, Jan 16 2023

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

a(n) ~ binomial(n^2,2*n).
a(n) ~ exp(2*n-2) * n^(2*n - 1/2) / (sqrt(Pi) * 2^(2*n+1)).
From Peter Bala, Jan 19 2023: (Start)
Conjectures: a(2^k) == 0 (mod 2^(k-1)) and a(3^k) == 0 (mod 3^(k+2)) for k >= 2; a(p^k) == 0 (mod p^(k+1)) for all primes p >= 5.
Let m be a positive integer. Similar recurrences may hold for the sequence whose n-th term is given by Sum_{k = 0..n} binomial(m*n*k, n+k). Cf. A099237. (End)

A096132 Triangle read by rows in which the r-th term of the n-th row is C(n^r,r*n), where r = 1 to n.

Original entry on oeis.org

1, 1, 1, 1, 84, 4686825, 1, 12870, 3284214703056, 10078751602022313874633200, 1, 3268760, 9064807833193439800, 25006639164538285144538957539300707000, 137658555538877668586244095134027016988748997970545868021484500, 1

Offset: 1

Amarnath Murthy, Jul 04 2004

			1
1 1
1 84 4686825
1 12870 3284214703056 = C(256,16) 10078751602022313874633200
1 3268760 9064807833193439800 25006639164538285144538957539300707000 ...
...

Cf. A096130, A096131.

Flatten[ Table[ Binomial[n^r, r*n], {n, 6}, {r, n}]] (* Robert G. Wilson v, Jul 08 2004 *)

Edited, corrected and extended by Robert G. Wilson v, Jul 08 2004

cp's OEIS Frontend

A226391 a(n) = Sum_{k=0..n} binomial(k*n, k).

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A361281 a(n) = n! * Sum_{k=0..n} binomial(n*k,n-k)/k!.

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A096130 Triangle read by rows: T(n,k) = binomial(k*n,n), 1 <= k <= n.

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A336513 a(n) = Sum_{i=1..n} Product_{j=(i-1)*n+1..i*n} j.

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A349470 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(k*n,n).

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A029848 a(n) = 1 + C(2*n,n) + C(3*n,n).

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GAP

Maple

Mathematica

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A275779 a(n) = (2^(n^2) - 1)/(1 - 1/2^n).

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Mathematica

A336513 a(n) = Sum_{i=1..n} Product_{j=(i-1)n+1..in} j.

A029848 a(n) = 1 + C(2n,n) + C(3n,n).