A076110 -id:A076110 - cp's OEIS frontend

A076111 Product of terms in n-th row of A076110.

1, 2, 15, 280, 9945, 576576, 49579075, 5925744000, 939536222625, 190787784140800, 48279601331512551, 14894665739501184000, 5502449072258318805625, 2397950328737212204032000

Offset: 0

Amarnath Murthy, Oct 09 2002

G. C. Greubel, Table of n, a(n) for n = 0..230

Cf. A002522, A064808, A076110.

List([0..15], n-> Product([1..n], j-> j*n+1) ); # G. C. Greubel, Mar 04 2020

[1] cat [&*[j*n+1: j in [1..n]]: n in [1..15]]; // G. C. Greubel, Mar 04 2020

seq( mul(j*n+1, j=1..n), n=0..15); # G. C. Greubel, Mar 04 2020

Table[Product[j*n+1, {j,n}], {n,0,15}] (* G. C. Greubel, Mar 04 2020 *)

A076111(n):=prod(1+n*k,k,1,n)$
makelist(A076111(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

vector(16, n, my(m=n-1); prod(j=1,m, j*m+1)) \\ G. C. Greubel, Mar 04 2020

[product(j*n+1 for j in (1..n)) for n in (0..15)] # G. C. Greubel, Mar 04 2020

a(n) = Prod_{k=1..n} (1+n*k). - Yalcin Aktar, Jul 14 2009
a(n) = n^n * Pochhammer(n, 1 + 1/n). - G. C. Greubel, Mar 04 2020
a(n) = A092985(n)*(n^2+1). - R. J. Mathar, Mar 30 2023

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

A064808 a(n) is the (n+1)st (n+2)-gonal number.

Original entry on oeis.org

1, 3, 9, 22, 45, 81, 133, 204, 297, 415, 561, 738, 949, 1197, 1485, 1816, 2193, 2619, 3097, 3630, 4221, 4873, 5589, 6372, 7225, 8151, 9153, 10234, 11397, 12645, 13981, 15408, 16929, 18547, 20265, 22086, 24013, 26049, 28197, 30460, 32841, 35343, 37969, 40722

Offset: 0

Floor van Lamoen, Oct 22 2001

Sum of n terms of the arithmetic progression with first term 1 and common difference n-1. - Amarnath Murthy, Aug 04 2005
a(n) is the sum of (n+1)-th row terms of triangle A144693. - Gary W. Adamson, Sep 19 2008
See also A131685(k) = smallest positive number m such that c(i) = m*(i^1 + 1)*(i^2 + 2)* ... *(i^k+ k) / k! takes integral values for all i>=0: For k=2, A131685(k)=1, which implies that this is a well-defined integer sequence. - Alexander R. Povolotsky, Apr 24 2015

Harry J. Smith, Table of n, a(n) for n = 0..1000
Justin Crum, Cyrus Cheng, David A. Ham, Lawrence Mitchell, Robert C. Kirby, Joshua A. Levine, and Andrew Gillette, Bringing Trimmed Serendipity Methods to Computational Practice in Firedrake, arXiv:2104.12986 [math.NA], 2021.
Index to divisibility sequences
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Main diagonal of A057145.
Row sums of A076110.
Cf. A144693. - Gary W. Adamson, Sep 19 2008
Cf. A000096, A002378, A006002, A006003.

[(n+1)*(n^2+2)/2 : n in [0..50]]; // Wesley Ivan Hurt, Feb 21 2015

A064808:=n->(n+1)*(n^2+2)/2: seq(A064808(n), n=0..50); # Wesley Ivan Hurt, Feb 21 2015

Table[(n + 1) (n^2 + 2)/2, {n, 0, 50}] (* Wesley Ivan Hurt, Feb 21 2015 *)

a(n) = { (n + 1)*(n^2 + 2)/2 } \\ Harry J. Smith, Sep 26 2009

a(n) = (n+1)*(n^2 + 2)/2.
From Paul Barry, Nov 18 2005: (Start)
a(n) = Sum_{k=0..n} Sum_{j=0..n} (k-(k-1)*C(0, j-k)).
a(n) = A006002(n) - A000096(n-2). (End)
G.f.: (1 - x + 3x^2)/(1 - x)^4. - R. J. Mathar, Jul 07 2009
a(n) = A006003(n+1) - A002378(n). - Rick L. Shepherd, Feb 21 2015
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Feb 21 2015
a(n) = A057145(n+2,n+1). - R. J. Mathar, Jul 28 2016

A162609 Triangle read by rows in which row n lists n terms, starting with 1, with gaps = n-2 between successive terms.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 5, 7, 1, 4, 7, 10, 13, 1, 5, 9, 13, 17, 21, 1, 6, 11, 16, 21, 26, 31, 1, 7, 13, 19, 25, 31, 37, 43, 1, 8, 15, 22, 29, 36, 43, 50, 57, 1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81

Offset: 1

Omar E. Pol, Jul 09 2009

Equals A081493 when first column is removed. - Georg Fischer, Jul 25 2023

			Triangle begins:
  1;
  1,  1;
  1,  2,  3;
  1,  3,  5,  7;
  1,  4,  7, 10, 13;
  1,  5,  9, 13, 17, 21;
  1,  6, 11, 16, 21, 26, 31;

Cf. A002061, A159797, A159798, A162610, A161611, A162612, A162613.

Cf. A060354 (row sums), A081493 (without first column).

Table[NestList[#+(n-2)&,1,n-1],{n,20}]//Flatten (* Harvey P. Dale, Oct 23 2017 *)

T(n,n) = A002061(n-1).

T(n,k) = A076110(n-1,k) = 1+(n-2)*(k-1). - R. J. Mathar, Mar 30 2023

A092985 a(n) is the product of the first n terms of an arithmetic progression with the first term 1 and common difference n.

Original entry on oeis.org

1, 1, 3, 28, 585, 22176, 1339975, 118514880, 14454403425, 2326680294400, 478015854767451, 122087424094272000, 37947924636264267625, 14105590169042424729600, 6178966019176767549393375, 3150334059785191453342744576, 1849556085478041490537172810625

Offset: 0

Amarnath Murthy, Mar 28 2004

We have the triangle (chopped versions of A076110, A162609)
  1;
  1 3;
  1 4  7;
  1 5  9 13;
  1 6 11 16 21;
  1 7 13 19 25 31;
...
Sequence contains the product of the terms of the rows.
a(n) = b(n-1) where b(n) = n^n*Gamma(n+1/n)/Gamma(1/n) and b(0) is limit n->0+ of b(n). - Gerald McGarvey, Nov 10 2007
Product of the entries in the first column of an n X n square array with elements 1..n^2 listed in increasing order by rows. - Wesley Ivan Hurt, Apr 02 2025

			a(5) = 1*6*11*16*21 = 22176.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A057237, A092987.

Main diagonal of A256268.

List([0..20], n-> Product([0..n-1], j-> j*n+1) ); # G. C. Greubel, Mar 04 2020

[1] cat [ (&*[j*n+1: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Mar 04 2020

a:= n-> mul(n*j+1, j=0..n-1):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 24 2015

Flatten[{1, Table[n^n * Pochhammer[1/n, n], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 05 2018 *)

vector(21, n, my(m=n-1); prod(j=0,m-1, j*m+1)) \\ G. C. Greubel, Mar 04 2020

[product(j*n+1 for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Mar 04 2020

a(n) = Product_{k=1..n} (1+(k-1)*n) = 1*(1+n)*(1+2n)*...*(n^2-n+1).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*n^(n-k). - Vladeta Jovovic, Jan 28 2005
a(n) = n! * [x^n] 1/(1 - n*x)^(1/n) for n > 0. - Ilya Gutkovskiy, Oct 05 2018
a(n) ~ sqrt(2*Pi) * n^(2*n - 3/2) / exp(n). - Vaclav Kotesovec, Oct 05 2018

More terms from Erich Friedman, Aug 08 2005

Offset corrected by Alois P. Heinz, Nov 24 2015

cp's OEIS Frontend

A076111 Product of terms in n-th row of A076110.

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A064808 a(n) is the (n+1)st (n+2)-gonal number.

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A162609 Triangle read by rows in which row n lists n terms, starting with 1, with gaps = n-2 between successive terms.

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A092985 a(n) is the product of the first n terms of an arithmetic progression with the first term 1 and common difference n.

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Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions