A002415 -id:A002415 - cp's OEIS frontend

A191662 a(n) = n! / A000034(n-1).

1, 1, 6, 12, 120, 360, 5040, 20160, 362880, 1814400, 39916800, 239500800, 6227020800, 43589145600, 1307674368000, 10461394944000, 355687428096000, 3201186852864000, 121645100408832000, 1216451004088320000, 51090942171709440000, 562000363888803840000

Offset: 1

Paul Curtz, Jun 10 2011

The a(n) are the denominators in the formulas of the k-dimensional square pyramidal numbers:
A005408 = (2*n+1)/1                       = 1, 3,  5,   7,   9, ... (k=1)
A000290 = (n^2)/1                         = 1, 4,  9,  16,  25, ... (k=2)
A000330 = n*(n+1)*(2*n+1)/6               = 1, 5, 14,  30,  55, ... (k=3)
A002415 = (n^2)*(n^2-1)/12                = 1, 6, 20,  50, 105, ... (k=4)
A005585 = n*(n+1)*(n+2)*(n+3)*(2*n+3)/120 = 1, 7, 27,  77, 182, ... (k=5)
A040977 = (n^2)*(n^2-1)*(n^2-4)/360       = 1, 8, 35, 112, 294, ... (k=6)
A050486 (k=7), A053347 (k=8), A054333 (k=9), A054334 (k=10), A057788 (k=11).
The first superdiagonal of this array appears in A029651. - Paul Curtz, Jul 04 2011
The general formula for the k-dimensional square pyramidal numbers is (2*n+k)*binomial(n+k-1,k-1)/k, k >= 1, n >= 0, see A097207. - Johannes W. Meijer, Jun 22 2011

Cf. A000142, A009445, A002674, A064680.
Cf. A000290, A000330, A002415, A005408, A005585, A029651, A040977, A050486, A053347, A054333, A054334, A057788.
Cf. A052612.

A191662:= proc(n): n!/A000034(n-1) end: A000034 := proc(n) op((n mod 2)+1, [1, 2]) ; end proc: seq(A191662(n),n=1..17); # Johannes W. Meijer, Jun 22 2011

Array[If[EvenQ[#],#!/2,#!]&,20] (* Harvey P. Dale, Mar 14 2014 *)

a(2*n-1) = (2*n-1)!, a(2*n) = (2*n)!/2.
a(n+1) = A064680(n+1) * a(n).
From Amiram Eldar, Jul 06 2022: (Start)
Sum_{n>=1} 1/a(n) = sinh(1) + 2*cosh(1) - 2.
Sum_{n>=1} (-1)^(n+1)/a(n) = sinh(1) - 2*cosh(1) + 2. (End)
D-finite with recurrence: a(n) - (n-1)*n*a(n-2) = 0 for n >= 3 with a(1)=a(2)=1. - Georg Fischer, Nov 25 2022
a(n) = A052612(n)/2 for n >= 1. - Alois P. Heinz, Sep 05 2023

More terms from Harvey P. Dale, Mar 14 2014

A208508 Triangle of coefficients of polynomials u(n,x) jointly generated with A208509; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 9, 1, 1, 16, 6, 1, 25, 20, 1, 1, 36, 50, 8, 1, 49, 105, 35, 1, 1, 64, 196, 112, 10, 1, 81, 336, 294, 54, 1, 1, 100, 540, 672, 210, 12, 1, 121, 825, 1386, 660, 77, 1, 1, 144, 1210, 2640, 1782, 352, 14, 1, 169, 1716, 4719, 4290, 1287, 104, 1, 1

Offset: 1

Clark Kimberling, Feb 27 2012

col 1:  A000012
col 2:  A000290 (squares)
col 3:  A002415
col 4:  A040977
col 5:  A054334
row sums, u(n,1): A083329

			First five rows:
1
1...1
1...4
1...9....1
1...16...6
First five polynomials u(n,x):
1
1 + x
1 + 4x
1 + 9x + x^2
1 + 16x + 6x^2

Cf. A208509.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208508 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208509 *)

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A212891 Rectangular array: (row n) = bc, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 6, 4, 20, 17, 9, 50, 46, 34, 16, 105, 100, 84, 57, 25, 196, 190, 170, 134, 86, 36, 336, 329, 305, 260, 196, 121, 49, 540, 532, 504, 450, 370, 270, 162, 64, 825, 816, 784, 721, 625, 500, 356, 209, 81, 1210, 1200, 1164, 1092, 980, 830, 650, 454, 262

Offset: 1

Clark Kimberling, Jun 16 2012

Principal diagonal:  A213436
Antidiagonal sums:  A024166
row 1,  (1,2,3,...)**(1,4,9,...):  A002415(k+1)
row 2,  (1,2,3,...)**(4,9,16,...):  k*(k^3 + 8*k^2 + 23*k + 16)/12
row 3,  (1,2,3,...)**(9,16,25,...):  k*(k^3 + 12*k^2 + 53*k + 42)/12
...
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....6....20....50....105....196...336
4....17...46....100...190....329...532
9....34...84....170...305....504...784
16...57...134...260...450....721...1092
25...86...196...370...625....980...1456
...
T(5,1) = (1)**(25) = 25
T(5,2) = (1,2)**(25,36) = 1*36+2*25 = 86
T(5,3) = (1,2,3)**(25,36,49) = 1*49+2*36+3*25 = 196

Cf. A213500.

b[n_] := n; c[n_] := n^2
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A212891 *)
d = Table[t[n, n], {n, 1, 40}] (* A213436 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A024166  *)

T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = n^2 - (2*n^2 - 2*n - 1)*x + ((n-1)^2)*x^2 and g(x) = (1 - x)^5.

A334881 Number of squares in 3-dimensional space whose four vertices have coordinates (x,y,z) in the set {1,...,n}.

Original entry on oeis.org

0, 0, 6, 54, 240, 810, 2274, 5304, 10752, 19992, 34854, 57774, 91200, 139338, 206394, 296832, 417120, 575556, 779238, 1037514, 1359792, 1760694, 2251362, 2845140, 3554976, 4404876, 5416278, 6605946, 7996896, 9621678, 11500962, 13667772, 16143552, 18973608, 22190406

Offset: 0

Peter Kagey, May 14 2020

nonn

a(n) >= 3*n*A002415(n).

			For n = 5, one such square has vertex set {(2,1,1), (5,4,1), (4,5,5), (1,2,5)}.

Zachary Kaplan, Table of n, a(n) for n = 0..100
Zachary Kaplan, Python program
Mathematics Stack Exchange user Olivier Massicot, How many squares in a three-dimensional n X n X n cartesian grid?

Cf. A002415 (squares in square grid), A098928 (cubes in cube grid).

Cf. A000332, A077435, A085582, A102698, A103158, A187452, A189412-A189418, A269747, A271910, A334581.

a(7)-a(12) from Pontus von Brömssen, May 15 2020
a(13)-a(20) from Peter Kagey, Jul 29 2020 via Mathematics Stack Exchange link
Terms a(21) and beyond from Zachary Kaplan, Sep 01 2020, via Mathematics Stack Exchange link

A338323 a(n) is the number of regular k-gons in three dimensions with all k vertices (x,y,z) in the set {1,2,...,n}^3.

Original entry on oeis.org

0, 0, 14, 138, 640, 2190, 6042, 13824, 28400, 53484, 94126, 156462, 248568, 380802, 564242, 813528, 1146472, 1581936, 2143878, 2857194, 3749240, 4854942, 6210442, 7856340, 9832056, 12194784, 15002678, 18312486, 22183672, 26693382, 31909362, 37916916, 44802728

Offset: 0

Peter Kagey, Oct 22 2020

nonn

The only regular polygons that can appear are equilateral triangles, squares, and regular hexagons.

			For the 3 X 3 X 3 grid, the a(3) = 138 regular polygons are A102698(3-1) = 80 triangles, A334881(3) = 54 squares, and A338322(3) = 4 hexagons.
An example of each shape, listed by the coordinates of their vertices:
Triangle: (1,2,1), (2,1,3), (3,3,2)
Square:   (1,1,1), (2,1,1), (2,2,1), (1,2,1)
Hexagon:  (1,1,2), (1,2,3), (2,1,1), (2,3,3), (3,2,1), (3,3,2)

Peter Kagey, Table of n, a(n) for n = 0..100
Code Golf Stack Exchange, Polygons in a cube
Burkard Polster, What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented, Mathologer video (2020).

Cf. A102698 (equilateral triangles), A334881 (squares), A338322 (regular hexagons).

The two-dimensional case is given by A002415.

a(n) = A102698(n-1) + A334881(n) + A338322(n) for n >= 2.

A072819 Variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -n for the first time.

Original entry on oeis.org

0, 0, 8, 48, 160, 400, 840, 1568, 2688, 4320, 6600, 9680, 13728, 18928, 25480, 33600, 43520, 55488, 69768, 86640, 106400, 129360, 155848, 186208, 220800, 260000, 304200, 353808, 409248, 470960, 539400, 615040, 698368, 789888, 890120, 999600

Offset: 0

Henry Bottomley, Jul 14 2002

			a(2)=8 since for a random walk with absorbing boundaries at +2 or -2, the probability of first reaching a boundary at time t=2 is 1/2, at t=4 is 1/4, at t=6 is 1/8, at t=8 is 1/16, etc., giving a mean of 2/2 + 4/4 + 6/8 + 8/16 + ... = 4 and a variance of 2^2/2 + 4^2/4 + 6^2/8 + 8^2/16 + ... - 4^2 = 24 - 16 = 8.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290 (i.e., n^2) for mean time. A072818(n)=sqrt(a(A001079(n))) attempts to identify the integer standard deviations.

[n^2*(n^2-1)*2/3: n in [0..40]]; // Vincenzo Librandi, Sep 14 2011

CoefficientList[Series[8 (1 + x) x^2/(1 - x)^5, {x, 0, 35}], x] (* Michael De Vlieger, Jul 02 2019 *)

a(n) = n^2*(n^2 - 1)*2/3 = 4*A008911(n) = 8*A002415(n) = A069971(n, n).
G.f.: 8*(1 + x)*x^2/(1 - x)^5. - Arkadiusz Wesolowski, Feb 08 2012
E.g.f.: 2*exp(x)*x^2*(6 + 6*x + x^2)/3. - Stefano Spezia, Dec 12 2021
a(n) = 2*n * A007290(n+1). - C.S. Elder, Jan 09 2024

A079034 Determinant of M(n), the n X n matrix defined by m(i,i) = 1, m(i,j) = i-j.

Original entry on oeis.org

1, 1, 2, 7, 21, 51, 106, 197, 337, 541, 826, 1211, 1717, 2367, 3186, 4201, 5441, 6937, 8722, 10831, 13301, 16171, 19482, 23277, 27601, 32501, 38026, 44227, 51157, 58871, 67426, 76881, 87297, 98737, 111266, 124951, 139861, 156067, 173642, 192661, 213201, 235341

Offset: 0

Benoit Cloitre, Feb 01 2003

Starting (1, 1, 2, 7, 21, 51, 106, ...), = Narayana transform (A001263) of [1, 0, 1, 0, 0, 0, ...]. - Gary W. Adamson, Jan 04 2008

In 2022, Han Wang and Zhi-Wei Sun provided a proof of the formula a(n) = 1 + n^2*(n^2-1)/12 via eigenvalues. See A355175 for my conjecture on det[(i-j)^2+d(i,j)]{1<=i,j<=n}, where d(i,j) is 1 or 0 according as i = j or not. - _Zhi-Wei Sun, Jun 28 2022

Han Wang and Zhi-Wei Sun, Evaluations of three determinants, arXiv:2206.12317 [math.NT], 2022.
Han Wang and Zhi-Wei Sun, Characteristic polynomials of the matrices with (j, k)-entry q^(j±k) + t, Bull. Australian Math. Soc. (2024). See references.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001263, A002415, A355175.

LinearRecurrence[{5,-10,10,-5,1},{1,2,7,21,51},50] (* Harvey P. Dale, Aug 17 2014 *)

a(n) = (n^4-n^2+12)/12; a(n) = A002415(n)+1.

G.f.: (x^4-3*x^3+7*x^2-4*x+1) / (1-x)^5. - Colin Barker, Jun 24 2013

a(0)=1 prepended by Alois P. Heinz, Oct 23 2024

A081283 An interleaved sequence of pyramidal and polygonal numbers.

Original entry on oeis.org

0, 1, 1, 5, 6, 16, 20, 40, 50, 85, 105, 161, 196, 280, 336, 456, 540, 705, 825, 1045, 1210, 1496, 1716, 2080, 2366, 2821, 3185, 3745, 4200, 4880, 5440, 6256, 6936, 7905, 8721, 9861, 10830, 12160, 13300, 14840, 16170, 17941, 19481, 21505, 23276, 25576, 27600

Offset: 0

Paul Barry, Mar 17 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A081282, A081284.

A081283:=n->(2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(2*n^2+2*(5+(-1)^n)*n+27-11*(-1)^n)/1536: seq(A081283(n), n=0..80); # Wesley Ivan Hurt, Apr 18 2017

CoefficientList[Series[x (1 + x^3) / ((1 - x) (1 - x^2)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 07 2013 *)

G.f.: x*(1+x^3)/((1-x)*(1-x^2)^4).
a(2*n) = A002415(n);  a(2*n+1) = A006007(n+1).
a(n) = (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(2*n^2+2*(5+(-1)^n)*n+27-11*(-1)^n)/1536. - Luce ETIENNE, Mar 11 2015

A177877 Triangle in which row n is derived from (1,2,3,...,n) dot (n,n-1,...,1) with additive carryovers.

Original entry on oeis.org

1, 2, 4, 3, 7, 10, 4, 10, 16, 20, 5, 13, 22, 30, 35, 6, 16, 28, 40, 50, 56, 7, 19, 34, 50, 65, 77, 84, 8, 22, 40, 60, 80, 98, 112, 120, 9, 25, 46, 70, 95, 119, 140, 156, 165, 10, 28, 52, 80, 110, 140, 168, 192, 210, 220

Offset: 0

Gary W. Adamson, Dec 13 2010

Carryovers (additive) are defined as "add current product to next product". For example: (1,2,3) dot (3,2,1) with carryovers = ((1*3=3), (2*2+3=7), (3*1+7=10)), so row 2 = (3, 7, 10).

			Row 2 = (3, 7, 10) = (1, 2, 3) dot (3, 2, 1) with carryovers, thus: (3 = 1*3; 7 = 2*2 + 3; 10 = 3*1 + 7).
First few rows of the array:
  1,   4,  10,  20,  35,...
  2,   7,  16,  30,  50,...
  3,  10,  22,  40,  65,...
  ...
Example: row 1 is obtained by adding (1, 3, 6, 10, 15,...) termwise to (1, 4, 10, 20, 35,...).
First few rows of the triangle:
  1;
  2, 4;
  3, 7, 10;
  4, 10, 16, 20;
  5, 13, 22, 30, 35;
  6, 16, 28, 40, 50, 56;
  7, 19, 34, 50, 65, 77, 84;
  8, 22, 40, 60, 80, 98, 112, 120;
  9, 25, 46, 70, 95, 119, 140, 156, 165;
  10, 28, 52, 80, 110, 140, 168, 192, 210, 220;
  ...

Cf. A002415 (row sums).

By rows, (1,2,3,...) dot (...3,2,1); add current product to next product.

As an array, row 0 = the tetrahedral numbers, (1, 4, 10, 20, 35,...). n-th row adds n*(1, 3, 6, 10, 15,...) termwise.

A202674 Symmetric matrix based on (1,3,5,7,9,...), by antidiagonals.

Original entry on oeis.org

1, 3, 3, 5, 10, 5, 7, 18, 18, 7, 9, 26, 35, 26, 9, 11, 34, 53, 53, 34, 11, 13, 42, 71, 84, 71, 42, 13, 15, 50, 89, 116, 116, 89, 50, 15, 17, 58, 107, 148, 165, 148, 107, 58, 17, 19, 66, 125, 180, 215, 215, 180, 125, 66, 19, 21, 74, 143, 212, 265, 286, 265, 212

Offset: 1

Clark Kimberling, Dec 22 2011

Let s=(1,3,5,7,9,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s.  Let T' be the transpose of T.  Then A202674 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A202675 for characteristic polynomials of principal submatrices of M.
...
row 1 (1,3,5,7,...) A005408
diagonal (1,10,35,84,...) A000447
antidiagonal sums (1,6,20,50,...) A002415

			Northwest corner:
1....3....5.....7.....9
3...10...18....26....34
5...18...35....53....71
7...26...53....84...116
9...34...71...116...165

Cf. A005408, A202675, A193722.

U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[2 k - 1, {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

cp's OEIS Frontend

A191662 a(n) = n! / A000034(n-1).

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A208508 Triangle of coefficients of polynomials u(n,x) jointly generated with A208509; see the Formula section.

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A212891 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

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A334881 Number of squares in 3-dimensional space whose four vertices have coordinates (x,y,z) in the set {1,...,n}.

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A338323 a(n) is the number of regular k-gons in three dimensions with all k vertices (x,y,z) in the set {1,2,...,n}^3.

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A072819 Variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -n for the first time.

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Magma

Mathematica

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A079034 Determinant of M(n), the n X n matrix defined by m(i,i) = 1, m(i,j) = i-j.

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Mathematica

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Extensions

A081283 An interleaved sequence of pyramidal and polygonal numbers.

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Links

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Programs

Maple

A212891 Rectangular array: (row n) = bc, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and = convolution.