A226391 -id:A226391 - cp's OEIS frontend

A096131 Sum of the terms of the n-th row of triangle pertaining to A096130.

1, 7, 105, 2386, 71890, 2695652, 120907185, 6312179764, 375971507406, 25160695768715, 1869031937691061, 152603843369288819, 13584174777196666630, 1309317592648179024666, 135850890740575408906465

Offset: 1

Amarnath Murthy, Jul 04 2004

nonn

The product of the terms of the n-th row is given by A034841.

Collection of partial binary matrices: 1 to n rows of length n and a total of n entries set to one in each partial matrix. - Olivier Gérard, Aug 08 2016

			From _Seiichi Manyama_, Aug 18 2018: (Start)
a(1) = (1/1!) * (1) = 1.
a(2) = (1/2!) * (1*2 + 3*4) = 7.
a(3) = (1/3!) * (1*2*3 + 4*5*6 + 7*8*9) = 105.
a(4) = (1/4!) * (1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16) = 2386. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..338

Cf. A014062, A096130, A034841, A007318, A226391, A167009, A167008, A167010, A072034, A086331, A349470.

List(List([1..20],n->List([1..n],k->Binomial(k*n,n))),Sum); # Muniru A Asiru, Aug 12 2018

A096130 := proc(n,k) binomial(k*n,n) ; end: A096131 := proc(n) local k; add( A096130(n,k),k=1..n) ; end: for n from 1 to 18 do printf("%d, ",A096131(n)) ; od ; # R. J. Mathar, Apr 30 2007
seq(add((binomial(n*k,n)), k=0..n), n=1..15); # Zerinvary Lajos, Sep 16 2007

Table[Sum[Binomial[k*n, n], {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jun 06 2013 *)

a(n) = sum(k=1, n, binomial(k*n, n)); \\ Michel Marcus, Aug 20 2018

a(n) = Sum_{k=1..n} binomial(k*n, n). - Reinhard Zumkeller, Jan 09 2005
a(n) = (1/n!) * Sum_{j=1..n} Product_{i=n*(j-1)+1..n*j} i. - Reinhard Zumkeller, Jan 09 2005 [corrected by Seiichi Manyama, Aug 17 2018]
a(n) ~ exp(1)/(exp(1)-1) * binomial(n^2,n). - Vaclav Kotesovec, Jun 06 2013

More terms from R. J. Mathar, Apr 30 2007

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

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

A295772 a(n) = Sum_{k=0..n} binomial((n-k)*k, k).

Original entry on oeis.org

1, 1, 2, 4, 11, 41, 189, 1020, 6277, 43262, 328963, 2727076, 24425913, 234743744, 2406904525, 26202132494, 301579542517, 3656552470482, 46555182556971, 620695577790512, 8644238847922949, 125472134647552497, 1894393648378487895, 29696659293381522674

Offset: 0

Vaclav Kotesovec, Nov 27 2017

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 41*x^5 + 189*x^6 + 1020*x^7 + 6277*x^8 + 43262*x^9 + 328963*x^10 + ...

Robert Israel, Table of n, a(n) for n = 0..508

Cf. A060539, A226391, A264409.

seq(add(binomial((n-k)*k,k),k=0..n),n=0..30); # Robert Israel, Nov 27 2017

Table[Sum[Binomial[(n-k)*k, k], {k, 0, n}], {n, 0, 30}]

log(a(n)) ~ n*(log(n) - log(log(n)) + (log(log(n)) - 1)/log(n)). - Vaclav Kotesovec, Jan 10 2023

G.f. A(x) = 1 + x*Sum_{n>=0} x^n/n! * ( d^n/dy^n (1+y)^n/(1 - x*(1+y)^n) ) evaluated at y = 0. - Paul D. Hanna, Nov 13 2024

A295773 a(n) = Sum_{k=0..n} binomial(k^2, k).

Original entry on oeis.org

1, 2, 8, 92, 1912, 55042, 2002834, 87903418, 4514068786, 265401903136, 17575711359576, 1294325676386112, 104913619501093500, 9281271920245432932, 889811788303594625412, 91895379599481072720852, 10170646981621794947354052, 1200909691326112843842751962

Offset: 0

Vaclav Kotesovec, Nov 27 2017

nonn

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

Cf. A014062, A226391, A295772, A359665.

[&+[Binomial(k^2, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Jan 10 2019

Table[Sum[Binomial[k^2, k], {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(k^2, k)); \\ Michel Marcus, Jan 10 2019

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

A349471 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(k*n,k).

Original entry on oeis.org

1, 0, 5, 71, 1625, 48699, 1815157, 80960200, 4205895521, 249447427145, 16631893722851, 1231521399730489, 100270564101729529, 8903719880410535595, 856322102196977446955, 88677383473792696758599, 9837660365763014667911553, 1163993530309417488368300653

Offset: 0

Seiichi Manyama, Nov 19 2021

nonn

Cf. A226391, A349470.

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

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

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

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));

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)

A383120 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k).

Original entry on oeis.org

1, 2, 11, 139, 2885, 82381, 2979565, 130203494, 6664589321, 390857822425, 25832193906761, 1899273577364197, 153741850998047053, 13585520026454056279, 1301210398133681268381, 134270617908678099820891, 14849785991790603714043921, 1752283118795349858851381297

Offset: 0

Ilya Gutkovskiy, Apr 17 2025

nonn

Cf. A001850, A014062, A026375, A188686, A226391, A359643, A378327, A383121.

Table[Sum[Binomial[n, k] Binomial[n k, k], {k, 0, n}], {n, 0, 17}]

a(n) = sum(k=0, n, binomial(n,k) * binomial(n*k,k)); \\ Michel Marcus, Apr 17 2025

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

a(n) ~ exp(n + exp(-1) - 1/2) * n^n / sqrt(2*Pi*n). - Vaclav Kotesovec, Apr 17 2025

cp's OEIS Frontend

A096131 Sum of the terms of the n-th row of triangle pertaining to A096130.

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

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A295772 a(n) = Sum_{k=0..n} binomial((n-k)*k, k).

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A295773 a(n) = Sum_{k=0..n} binomial(k^2, k).

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A349471 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(k*n,k).

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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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PARI

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

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Maple

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A383120 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k).

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