A162990 -id:A162990 - cp's OEIS frontend

A162995 A scaled version of triangle A162990.

1, 3, 1, 12, 4, 1, 60, 20, 5, 1, 360, 120, 30, 6, 1, 2520, 840, 210, 42, 7, 1, 20160, 6720, 1680, 336, 56, 8, 1, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1

Offset: 1

Johannes W. Meijer, Jul 27 2009

We get this scaled version of triangle A162990 by dividing the coefficients in the left hand columns by their 'top-values' and then taking the square root.
T(n,k) = A173333(n+1,k+1), 1 <= k <= n. - Reinhard Zumkeller, Feb 19 2010
T(n,k) = A094587(n+1,k+1), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012

			The first few rows of the triangle are:
[1]
[3, 1]
[12, 4, 1]
[60, 20, 5, 1]

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A094587.
A056542(n) equals the row sums for n>=1.
A001710, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431 are related to the left hand columns.
A000012, A009056, A002378, A007531, A052762, A052787, A053625 and A159083 are related to the right hand columns.

a162995 n k = a162995_tabl !! (n-1) !! (k-1)
a162995_row n = a162995_tabl !! (n-1)
a162995_tabl = map fst $ iterate f ([1], 3)
   where f (row, i) = (map (* i) row ++ [1], i + 1)
-- Reinhard Zumkeller, Jul 04 2012

a := proc(n, m): (n+1)!/(m+1)! end: seq(seq(a(n, m), m=1..n), n=1..9); # Johannes W. Meijer, revised Nov 23 2012

Table[(n+1)!/(m+1)!, {n, 10}, {m, n}] (* Paolo Xausa, Mar 31 2024 *)

a(n,m) = (n+1)!/(m+1)! for n = 1, 2, 3, ..., and m = 1, 2, ..., n.

A162993 The second left hand column of triangle A162990.

Original entry on oeis.org

9, 144, 3600, 129600, 6350400, 406425600, 32920473600, 3292047360000, 398337730560000, 57360633200640000, 9693947010908160000, 1900013614137999360000, 427503063181049856000000, 109440784174348763136000000, 31628386626386792546304000000

Offset: 2

Johannes W. Meijer, Jul 21 2009

A001044 and A162994 are two other left hand columns.

A001710(n+1) equals the square root of a(n).

Array[((#+1)!/2)^2 &, 20, 2] (* Paolo Xausa, Apr 01 2024 *)

a(n) = ((n+1)!/2)^2 for n = 2, 3, ... .

a(n) = A001710(n+1)^2.

A162991 The first right hand column of triangle A162990.

Original entry on oeis.org

4, 9, 64, 900, 20736, 705600, 33177600, 2057529600, 162570240000, 15933509222400, 1896219279360000, 269276305858560000, 44970736429301760000, 8724552309817344000000, 1945613940877311344640000

Offset: 1

Johannes W. Meijer, Jul 21 2009

A001048(n) equals the square root of a(n).

A162992 is the second right hand column of triangle A162990.

Array[((#+1)!/#)^2 &, 20] (* Paolo Xausa, Mar 31 2024 *)

a(n) = ((n+1)!/n)^2 for n = 1, 2, 3, ... .

A162992 The second right hand column of triangle A162990.

Original entry on oeis.org

36, 144, 1600, 32400, 1016064, 45158400, 2687385600, 205752960000, 19670999040000, 2294425328025600, 320461058211840000, 52778155948277760000, 10118415696592896000000, 2233485391313240064000000

Offset: 2

Johannes W. Meijer, Jul 21 2009

A052747(n+1) equals the square root of a(n).

A162991 is the first right hand column of triangle A162990.

Array[((#+1)!/(#-1))^2 &, 20, 2] (* Paolo Xausa, Apr 01 2024 *)

a(n) = ((n+1)!/(n-1))^2 for n = 2, 3, ... .

a(n) = A052747(n+1)^2.

A162994 The third left hand column of triangle A162990.

Original entry on oeis.org

64, 1600, 57600, 2822400, 180633600, 14631321600, 1463132160000, 177038991360000, 25493614755840000, 4308420893736960000, 844450495172444160000, 190001361413799936000000, 48640348521932783616000000, 14057060722838574465024000000

Offset: 3

Johannes W. Meijer, Jul 21 2009

A001044 and A162993 are two other left hand columns.

A002301(n+1) equals the square root of a(n).

Array[((#+1)!/3)^2 &, 20, 3] (* Paolo Xausa, Apr 01 2024 *)

a(n) = ((n+1)!/3)^2 for n = 3, 4, 5, ... .

a(n) = A002301(n+1)^2.

A002378 Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).

Original entry on oeis.org

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550

Offset: 0

N. J. A. Sloane

4*a(n) + 1 are the odd squares A016754(n).
The word "pronic" (used by Dickson) is incorrect. - Michael Somos
According to the 2nd edition of Webster, the correct word is "promic". - R. K. Guy
a(n) is the number of minimal vectors in the root lattice A_n (see Conway and Sloane, p. 109).
Let M_n denote the n X n matrix M_n(i, j) = (i + j); then the characteristic polynomial of M_n is x^(n-2) * (x^2 - a(n)*x - A002415(n)). - Benoit Cloitre, Nov 09 2002
The greatest LCM of all pairs (j, k) for j < k <= n for n > 1. - Robert G. Wilson v, Jun 19 2004
First differences are a(n+1) - a(n) = 2*n + 2 = 2, 4, 6, ... (while first differences of the squares are (n+1)^2 - n^2 = 2*n + 1 = 1, 3, 5, ...). - Alexandre Wajnberg, Dec 29 2005
25 appended to these numbers corresponds to squares of numbers ending in 5 (i.e., to squares of A017329). - Lekraj Beedassy, Mar 24 2006
A rapid (mental) multiplication/factorization technique -- a generalization of Lekraj Beedassy's comment: For all bases b >= 2 and positive integers n, c, d, k with c + d = b^k, we have (n*b^k + c)*(n*b^k + d) = a(n)*b^(2*k) + c*d. Thus the last 2*k base-b digits of the product are exactly those of c*d -- including leading 0(s) as necessary -- with the preceding base-b digit(s) the same as a(n)'s. Examples: In decimal, 113*117 = 13221 (as n = 11, b = 10 = 3 + 7, k = 1, 3*7 = 21, and a(11) = 132); in octal, 61*67 = 5207 (52 is a(6) in octal). In particular, for even b = 2*m (m > 0) and c = d = m, such a product is a square of this type. Decimal factoring: 5609 is immediately seen to be 71*79. Likewise, 120099 = 301*399 (k = 2 here) and 99990000001996 = 9999002*9999998 (k = 3). - Rick L. Shepherd, Jul 24 2021
Number of circular binary words of length n + 1 having exactly one occurrence of 01. Example: a(2) = 6 because we have 001, 010, 011, 100, 101 and 110. Column 1 of A119462. - Emeric Deutsch, May 21 2006
The sequence of iterated square roots sqrt(N + sqrt(N + ...)) has for N = 1, 2, ... the limit (1 + sqrt(1 + 4*N))/2. For N = a(n) this limit is n + 1, n = 1, 2, .... For all other numbers N, N >= 1, this limit is not a natural number. Examples: n = 1, a(1) = 2: sqrt(2 + sqrt(2 + ...)) = 1 + 1 = 2; n = 2, a(2) = 6: sqrt(6 + sqrt(6 + ...)) = 1 + 2 = 3. - Wolfdieter Lang, May 05 2006
Nonsquare integers m divisible by ceiling(sqrt(m)), except for m = 0. - Max Alekseyev, Nov 27 2006
The number of off-diagonal elements of an (n + 1) X (n + 1) matrix. - Artur Jasinski, Jan 11 2007
a(n) is equal to the number of functions f:{1, 2} -> {1, 2, ..., n + 1} such that for a fixed x in {1, 2} and a fixed y in {1, 2, ..., n + 1} we have f(x) <> y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007
Numbers m >= 0 such that round(sqrt(m+1)) - round(sqrt(m)) = 1. - Hieronymus Fischer, Aug 06 2007
Numbers m >= 0 such that ceiling(2*sqrt(m+1)) - 1 = 1 + floor(2*sqrt(m)). - Hieronymus Fischer, Aug 06 2007
Numbers m >= 0 such that fract(sqrt(m+1)) > 1/2 and fract(sqrt(m)) < 1/2 where fract(x) is the fractional part (fract(x) = x - floor(x), x >= 0). - Hieronymus Fischer, Aug 06 2007
X values of solutions to the equation 4*X^3 + X^2 = Y^2. To find Y values: b(n) = n(n+1)(2n+1). - Mohamed Bouhamida, Nov 06 2007
Nonvanishing diagonal of A132792, the infinitesimal Lah matrix, so "generalized factorials" composed of a(n) are given by the elements of the Lah matrix, unsigned A111596, e.g., a(1)*a(2)*a(3) / 3! = -A111596(4,1) = 24. - Tom Copeland, Nov 20 2007
If Y is a 2-subset of an n-set X then, for n >= 2, a(n-2) is the number of 2-subsets and 3-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
a(n) coincides with the vertex of a parabola of even width in the Redheffer matrix, directed toward zero. An integer p is prime if and only if for all integer k, the parabola y = kx - x^2 has no integer solution with 1 < x < k when y = p; a(n) corresponds to odd k. - Reikku Kulon, Nov 30 2008
The third differences of certain values of the hypergeometric function 3F2 lead to the squares of the oblong numbers i.e., 3F2([1, n + 1, n + 1], [n + 2, n + 2], z = 1) - 3*3F2([1, n + 2, n + 2], [n + 3, n + 3], z = 1) + 3*3F2([1, n + 3, n + 3], [n + 4, n + 4], z = 1) - 3F2([1, n + 4, n + 4], [n + 5, n + 5], z = 1) = (1/((n+2)*(n+3)))^2 for n = -1, 0, 1, 2, ... . See also A162990. - Johannes W. Meijer, Jul 21 2009
Generalized factorials, [a.(n!)] = a(n)*a(n-1)*...*a(0) = A010790(n), with a(0) = 1 are related to A001263. - Tom Copeland, Sep 21 2011
For n > 1, a(n) is the number of functions f:{1, 2} -> {1, ..., n + 2} where f(1) > 1 and f(2) > 2. Note that there are n + 1 possible values for f(1) and n possible values for f(2). For example, a(3) = 12 since there are 12 functions f from {1, 2} to {1, 2, 3, 4, 5} with f(1) > 1 and f(2) > 2. - Dennis P. Walsh, Dec 24 2011
a(n) gives the number of (n + 1) X (n + 1) symmetric (0, 1)-matrices containing two ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012
a(n) is the number of positions of a domino in a rectangled triangular board with both legs equal to n + 1. - César Eliud Lozada, Sep 26 2012
a(n) is the number of ordered pairs (x, y) in [n+2] X [n+2] with |x-y| > 1. - Dennis P. Walsh, Nov 27 2012
a(n) is the number of injective functions from {1, 2} into {1, 2, ..., n + 1}. - Dennis P. Walsh, Nov 27 2012
a(n) is the sum of the positive differences of the partition parts of 2n + 2 into exactly two parts (see example). - Wesley Ivan Hurt, Jun 02 2013
a(n)/a(n-1) is asymptotic to e^(2/n). - Richard R. Forberg, Jun 22 2013
Number of positive roots in the root system of type D_{n + 1} (for n > 2). - Tom Edgar, Nov 05 2013
Number of roots in the root system of type A_n (for n > 0). - Tom Edgar, Nov 05 2013
From Felix P. Muga II, Mar 18 2014: (Start)
a(m), for m >= 1, are the only positive integer values t for which the Binet-de Moivre formula for the recurrence b(n) = b(n-1) + t*b(n-2) with b(0) = 0 and b(1) = 1 has a root of a square. PROOF (as suggested by Wolfdieter Lang, Mar 26 2014): The sqrt(1 + 4t) appearing in the zeros r1 and r2 of the characteristic equation is (a positive) integer for positive integer t precisely if 4t + 1 = (2m + 1)^2, that is t = a(m), m >= 1. Thus, the characteristic roots are integers: r1 = m + 1 and r2 = -m.
Let m > 1 be an integer. If b(n) = b(n-1) + a(m)*b(n-2), n >= 2, b(0) = 0, b(1) = 1, then lim_{n->oo} b(n+1)/b(n) = m + 1. (End)
Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices (chromatic polynomial) of the complete graphs (here simply K_2). - Tom Copeland, Apr 05 2014
The set of integers k for which k + sqrt(k + sqrt(k + sqrt(k + sqrt(k + ...) ... is an integer. - Leslie Koller, Apr 11 2014
a(n-1) is the largest number k such that (n*k)/(n+k) is an integer. - Derek Orr, May 22 2014
Number of ways to place a domino and a singleton on a strip of length n - 2. - Ralf Stephan, Jun 09 2014
With offset 1, this appears to give the maximal number of crossings between n nonconcentric circles of equal radius. - Felix Fröhlich, Jul 14 2014
For n > 1, the harmonic mean of the n values a(1) to a(n) is n + 1. The lowest infinite sequence of increasing positive integers whose cumulative harmonic mean is integral. - Ian Duff, Feb 01 2015
a(n) is the maximum number of queens of one color that can coexist without attacking one queen of the opponent's color on an (n+2) X (n+2) chessboard. The lone queen can be placed in any position on the perimeter of the board. - Bob Selcoe, Feb 07 2015
With a(0) = 1, a(n-1) is the smallest positive number not in the sequence such that Sum_{i = 1..n} 1/a(i-1) has a denominator equal to n. - Derek Orr, Jun 17 2015
The positive members of this sequence are a proper subsequence of the so-called 1-happy couple products A007969. See the W. Lang link there, eq. (4), with Y_0 = 1, with a table at the end. - Wolfdieter Lang, Sep 19 2015
For n > 0, a(n) is the reciprocal of the area bounded above by y = x^(n-1) and below by y = x^n for x in the interval [0, 1]. Summing all such areas visually demonstrates the formula below giving Sum_{n >= 1} 1/a(n) = 1. - Rick L. Shepherd, Oct 26 2015
It appears that, except for a(0) = 0, this is the set of positive integers n such that x*floor(x) = n has no solution. (For example, to get 3, take x = -3/2.) - Melvin Peralta, Apr 14 2016
If two independent real random variables, x and y, are distributed according to the same exponential distribution: pdf(x) = lambda * exp(-lambda * x), lambda > 0, then the probability that n - 1 <= x/y < n is given by 1/a(n). - Andres Cicuttin, Dec 03 2016
a(n) is equal to the sum of all possible differences between n different pairs of consecutive odd numbers (see example). - Miquel Cerda, Dec 04 2016
a(n+1) is the dimension of the space of vector fields in the plane with polynomial coefficients up to order n. - Martin Licht, Dec 04 2016
It appears that a(n) + 3 is the area of the largest possible pond in a square (A268311). - Craig Knecht, May 04 2017
Also the number of 3-cycles in the (n+3)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017
Also the Wiener index of the (n+2)-wheel graph. - Eric W. Weisstein, Sep 08 2017
The left edge of a Floyd's triangle that consists of even numbers: 0; 2, 4; 6, 8, 10; 12, 14, 16, 18; 20, 22, 24, 26, 28; ... giving 0, 2, 6, 12, 20, ... The right edge generates A028552. - Waldemar Puszkarz, Feb 02 2018
a(n+1) is the order of rowmotion on a poset obtained by adjoining a unique minimal (or maximal) element to a disjoint union of at least two chains of n elements. - Nick Mayers, Jun 01 2018
From Juhani Heino, Feb 05 2019: (Start)
For n > 0, 1/a(n) = n/(n+1) - (n-1)/n.
For example, 1/6 = 2/3 - 1/2; 1/12 = 3/4 - 2/3.
Corollary of this:
Take 1/2 pill.
Next day, take 1/6 pill. 1/2 + 1/6 = 2/3, so your daily average is 1/3.
Next day, take 1/12 pill. 2/3 + 1/12 = 3/4, so your daily average is 1/4.
And so on. (End)
From Bernard Schott, May 22 2020: (Start)
For an oblong number m >= 6 there exists a Euclidean division m = d*q + r with q < r < d which are in geometric progression, in this order, with a common integer ratio b. For b >= 2 and q >= 1, the Euclidean division is m = qb*(qb+1) = qb^2 * q + qb where (q, qb, qb^2) are in geometric progression.
Some examples with distinct ratios and quotients:
   6 | 4            30 | 25          42 | 18
     -----             -----            -----
   2 | 1      ,      5 |  1   ,       6 |  2    ,
and also:
  42 | 12          420 | 100
     -----             -----
   6 |  3     ,     20 |   4  .
Some oblong numbers also satisfy a Euclidean division m = d*q + r with q < r < d that are in geometric progression in this order but with a common noninteger ratio b > 1 (see A335064). (End)
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {2, 2n}]. For n=1, this collapses to [1; {2}]. - Magus K. Chu, Sep 09 2022
a(n-2) is the maximum irregularity over all trees with n vertices. The extremal graphs are stars. (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023
For n > 0, number of diagonals in a regular 2*(n+1)-gon that are not parallel to any edge (cf. A367204). - Paolo Xausa, Mar 30 2024
a(n-1) is the maximum Zagreb index over all trees with n vertices. The extremal graphs are stars. (The Zagreb index of a graph is the sum of the squares of the degrees over all vertices of the graph.) - Allan Bickle, Apr 11 2024
For n >= 1, a(n) is the determinant of the distance matrix of a cycle graph on 2*n + 1 vertices (if the length of the cycle is even such a determinant is zero). - Miquel A. Fiol, Aug 20 2024
For n > 1, the continued fraction expansion of sqrt(16*a(n)) is [2n+1; {1, 2n-1, 1, 8n+2}]. - Magus K. Chu, Nov 20 2024
For n>=2, a(n) is the number of faces on a n+1-zone rhombic zonohedron. Each pair of a collection of great circles on a sphere intersects at two points, so there are 2*binomial(n+1,2) intersections. The dual of the implied polyhedron is a rhombic zonohedron, its faces corresponding to the intersections. - Shel Kaphan, Aug 12 2025

			a(3) = 12, since 2(3)+2 = 8 has 4 partitions with exactly two parts: (7,1), (6,2), (5,3), (4,4). Taking the positive differences of the parts in each partition and adding, we get: 6 + 4 + 2 + 0 = 12. - _Wesley Ivan Hurt_, Jun 02 2013
G.f. = 2*x + 6*x^2 + 12*x^3 + 20*x^4 + 30*x^5 + 42*x^6 + 56*x^7 + ... - _Michael Somos_, May 22 2014
From _Miquel Cerda_, Dec 04 2016: (Start)
a(1) = 2, since 45-43 = 2;
a(2) = 6, since 47-45 = 2 and 47-43 = 4, then 2+4 = 6;
a(3) = 12, since 49-47 = 2, 49-45 = 4, and 49-43 = 6, then 2+4+6 = 12. (End)

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Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle.
Eric Weisstein's World of Mathematics, Wheel Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Wikipedia, Pronic number.
Wolfram Research, Hypergeometric Function 3F2, The Wolfram Functions site.
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Partial sums of A005843 (even numbers). Twice triangular numbers (A000217).
1/beta(n, 2) in A061928.
Cf. A035106, A087811, A119462, A127235, A049598, A124080, A033996, A028896, A046092, A000217, A005563, A046092, A001082, A059300, A059297, A059298, A166373, A002943 (bisection), A002939 (bisection), A078358 (complement).
A036689 and A036690 are subsequences. Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488. - Bruno Berselli, Jun 10 2013
Row n=2 of A185651.
Cf. A007745, A169810, A213541, A005369 (characteristic function).
Cf. A281026. - Bruno Berselli, Jan 16 2017
Cf. A045943 (4-cycles in triangular honeycomb acute knight graph), A028896 (5-cycles), A152773 (6-cycles).
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
A335064 is a subsequence.
Second column of A003506.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).
Cf. A347213 (Dgf at s=4).
Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).
Cf. A016742, A049450.

a002378 n = n * (n + 1)
a002378_list = zipWith (*) [0..] [1..]
-- Reinhard Zumkeller, Aug 27 2012, Oct 12 2011

[n*(n+1) : n in [0..100]]; // Wesley Ivan Hurt, Oct 26 2015

A002378 := proc(n)
        n*(n+1) ;
end proc:
seq(A002378(n),n=0..100) ;

Table[n(n + 1), {n, 0, 50}] (* Robert G. Wilson v, Jun 19 2004 *)
oblongQ[n_] := IntegerQ @ Sqrt[4 n + 1]; Select[Range[0, 2600], oblongQ] (* Robert G. Wilson v, Sep 29 2011 *)
2 Accumulate[Range[0, 50]] (* Harvey P. Dale, Nov 11 2011 *)
LinearRecurrence[{3, -3, 1}, {2, 6, 12}, {0, 20}] (* Eric W. Weisstein, Jul 27 2017 *)

PARI
```
{a(n) = n*(n+1)};
```

concat(0, Vec(2*x/(1-x)^3 + O(x^100))) \\ Altug Alkan, Oct 26 2015

is(n)=my(m=sqrtint(n)); m*(m+1)==n \\ Charles R Greathouse IV, Nov 01 2018

is(n)=issquare(4*n+1) \\ Charles R Greathouse IV, Mar 16 2022

def a(n): return n*(n+1)
print([a(n) for n in range(51)]) # Michael S. Branicky, Jan 13 2022

(2 to 100 by 2).scanLeft(0)( + ) // Alonso del Arte, Sep 12 2019

G.f.: 2*x/(1-x)^3. - Simon Plouffe in his 1992 dissertation.
a(n) = a(n-1) + 2*n, a(0) = 0.
Sum_{n >= 1} a(n) = n*(n+1)*(n+2)/3 (cf. A007290, partial sums).
Sum_{n >= 1} 1/a(n) = 1. (Cf. Tijdeman)
Sum_{n >= 1} (-1)^(n+1)/a(n) = log(4) - 1 = A016627 - 1 [Jolley eq (235)].
1 = 1/2 + Sum_{n >= 1} 1/(2*a(n)) = 1/2 + 1/4 + 1/12 + 1/24 + 1/40 + 1/60 + ... with partial sums: 1/2, 3/4, 5/6, 7/8, 9/10, 11/12, 13/14, ... - Gary W. Adamson, Jun 16 2003
a(n)*a(n+1) = a(n*(n+2)); e.g., a(3)*a(4) = 12*20 = 240 = a(3*5). - Charlie Marion, Dec 29 2003
Sum_{k = 1..n} 1/a(k) = n/(n+1). - Robert G. Wilson v, Feb 04 2005
a(n) = A046092(n)/2. - Zerinvary Lajos, Jan 08 2006
Log 2 = Sum_{n >= 0} 1/a(2n+1) = 1/2 + 1/12 + 1/30 + 1/56 + 1/90 + ... = (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + ... = Sum_{n >= 0} (-1)^n/(n+1) = A002162. - Gary W. Adamson, Jun 22 2003
a(n) = A110660(2*n). - N. J. A. Sloane, Sep 21 2005
a(n-1) = n^2 - n = A000290(n) - A000027(n) for n >= 1. a(n) is the inverse (frequency distribution) sequence of A000194(n). - Mohammad K. Azarian, Jul 26 2007
(2, 6, 12, 20, 30, ...) = binomial transform of (2, 4, 2). - Gary W. Adamson, Nov 28 2007
a(n) = 2*Sum_{i=0..n} i = 2*A000217(n). - Artur Jasinski, Jan 09 2007, and Omar E. Pol, May 14 2008
a(n) = A006503(n) - A000292(n). - Reinhard Zumkeller, Sep 24 2008
a(n) = A061037(4*n) = (n+1/2)^2 - 1/4 = ((2n+1)^2 - 1)/4 = (A005408(n)^2 - 1)/4. - Paul Curtz, Oct 03 2008 and Klaus Purath, Jan 13 2022
a(0) = 0, a(n) = a(n-1) + 1 + floor(x), where x is the minimal positive solution to fract(sqrt(a(n-1) + 1 + x)) = 1/2. - Hieronymus Fischer, Dec 31 2008
E.g.f.: (x+2)*x*exp(x). - Geoffrey Critzer, Feb 06 2009
Product_{i >= 2} (1-1/a(i)) = -2*sin(Pi*A001622)/Pi = -2*sin(A094886)/A000796 = 2*A146481. - R. J. Mathar, Mar 12 2009, Mar 15 2009
E.g.f.: ((-x+1)*log(-x+1)+x)/x^2 also Integral_{x = 0..1} ((-x+1)*log(-x+1) + x)/x^2 = zeta(2) - 1. - Stephen Crowley, Jul 11 2009
a(A007018(n)) = A007018(n+1), i.e., A007018(n+1) = A007018(n)-th oblong numbers. - Jaroslav Krizek, Sep 13 2009
a(n) = floor((n + 1/2)^2). a(n) = A035608(n) + A004526(n+1). - Reinhard Zumkeller, Jan 27 2010
a(n) = 2*(2*A006578(n) - A035608(n)). - Reinhard Zumkeller, Feb 07 2010
a(n-1) = floor(n^5/(n^3 + n^2 + 1)). - Gary Detlefs, Feb 11 2010
For n > 1: a(n) = A173333(n+1, n-1). - Reinhard Zumkeller, Feb 19 2010
a(n) = A004202(A000217(n)). - Reinhard Zumkeller, Feb 12 2011
a(n) = A188652(2*n+1) + 1. - Reinhard Zumkeller, Apr 13 2011
For n > 0 a(n) = 1/(Integral_{x=0..Pi/2} 2*(sin(x))^(2*n-1)*(cos(x))^3). - Francesco Daddi, Aug 02 2011
a(n) = A002061(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(0) = 0, a(n) = A005408(A034856(n)) - A005408(n-1). - Ivan N. Ianakiev, Dec 06 2012
a(n) = A005408(A000096(n)) - A005408(n). - Ivan N. Ianakiev, Dec 07 2012
a(n) = A001318(n) + A085787(n). - Omar E. Pol, Jan 11 2013
Sum_{n >= 1} 1/(a(n))^(2s) = Sum_{t = 1..2*s} binomial(4*s - t - 1, 2*s - 1) * ( (1 + (-1)^t)*zeta(t) - 1). See Arxiv:1301.6293. - R. J. Mathar, Feb 03 2013
a(n)^2 + a(n+1)^2 = 2 * a((n+1)^2), for n > 0. - Ivan N. Ianakiev, Apr 08 2013
a(n) = floor(n^2 * e^(1/n)) and a(n-1) = floor(n^2 / e^(1/n)). - Richard R. Forberg, Jun 22 2013
a(n) = 2*C(n+1, 2), for n >= 0. - Felix P. Muga II, Mar 11 2014
A005369(a(n)) = 1. - Reinhard Zumkeller, Jul 05 2014
Binomial transform of [0, 2, 2, 0, 0, 0, ...]. - Alois P. Heinz, Mar 10 2015
a(2n) = A002943(n) for n >= 0, a(2n-1) = A002939(n) for n >= 1. - M. F. Hasler, Oct 11 2015
For n > 0, a(n) = 1/(Integral_{x=0..1} (x^(n-1) - x^n) dx). - Rick L. Shepherd, Oct 26 2015
a(n) = A005902(n) - A007588(n). - Peter M. Chema, Jan 09 2016
For n > 0, a(n) = lim_{m -> oo} (1/m)*1/(Sum_{i=m*n..m*(n+1)} 1/i^2), with error of ~1/m. - Richard R. Forberg, Jul 27 2016
From Ilya Gutkovskiy, Jul 28 2016: (Start)
Dirichlet g.f.: zeta(s-2) + zeta(s-1).
Convolution of nonnegative integers (A001477) and constant sequence (A007395).
Sum_{n >= 0} a(n)/n! = 3*exp(1). (End)
From Charlie Marion, Mar 06 2020: (Start)
a(n)*a(n+2k-1) + (n+k)^2 = ((2n+1)*k + n^2)^2.
a(n)*a(n+2k) + k^2 = ((2n+1)*k + a(n))^2. (End)
Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)/Pi. - Amiram Eldar, Jan 20 2021
A generalization of the Dec 29 2003 formula, a(n)*a(n+1) = a(n*(n+2)), follows. a(n)*a(n+k) = a(n*(n+k+1)) + (k-1)*n*(n+k+1). - Charlie Marion, Jan 02 2023
a(n) = A016742(n) - A049450(n). - Leo Tavares, Mar 15 2025

Additional comments from Michael Somos

Comment and cross-reference added by Christopher Hunt Gribble, Oct 13 2009

A001048 a(n) = n! + (n-1)!.

Original entry on oeis.org

2, 3, 8, 30, 144, 840, 5760, 45360, 403200, 3991680, 43545600, 518918400, 6706022400, 93405312000, 1394852659200, 22230464256000, 376610217984000, 6758061133824000, 128047474114560000, 2554547108585472000, 53523844179886080000, 1175091669949317120000

Offset: 1

N. J. A. Sloane, R. K. Guy

Number of {12, 12*, 1*2, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.
a(n) is the hook product of the shape (n, 1). - Emeric Deutsch, May 13 2004
From Jaume Oliver Lafont, Dec 01 2009: (Start)
(1+(x-1)*exp(x))/x = Sum_{k >= 1} x^k/a(k).
Setting x = 1 yields Sum_{k >= 1} 1/a(k) = 1. [Jolley eq 302] (End)
With regard to the comment by Jaume Oliver Lafont: P(n) = 1/a(n) is a probability distribution, with all values given as unit fractions. This distribution is connected to the Irwin-Hall distribution: Consider successively drawn random numbers, uniformly distributed in [0,1]. 1/a(n) is the probability for the sum of the random numbers exceeding 1 exactly with the (n+1)-th summand. P(n) has mean e-1 and variance 3e-e^2. From this we get e as the expected number of summands. - Manfred Boergens, May 20 2024
For n >= 2, a(n) is the size of the largest conjugacy class of the symmetric group on n + 1 letters. Equivalently, the maximum entry in each row of A036039. - Geoffrey Critzer, May 19 2013
In factorial base representation (A007623) the terms are written as: 10, 11, 110, 1100, 11000, 110000, ... From a(2) = 3 = "11" onward each term begins always with two 1's, followed by n-2 zeros. - Antti Karttunen, Sep 24 2016
e is approximately a(n)/A000255(n-1) for large n. - Dale Gerdemann, Jul 26 2019
a(n) is the number of permutations of [n+1] in which all the elements of [n] are cycle-mates, that is, 1,..,n are all in the same cycle. This result is readily shown after noting that the elements of [n] can be members of a n-cycle or an (n+1)-cycle. Hence a(n)=(n-1)!+n!. See an example below. - Dennis P. Walsh, May 24 2020

			For n=3, a(3) counts the 8 permutations of [4] with 1,2, and 3 all in the same cycle, namely, (1 2 3)(4), (1 3 2)(4), (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 2 4 3), (1 4 2 3), and (1 4 3 2). - _Dennis P. Walsh_, May 24 2020

L. B. W. Jolley, Summation of Series, Dover, 1961.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..100
Barry Balof and Helen Jenne, Tilings, Continued Fractions, Derangements, Scramblings, and e, Journal of Integer Sequences, Vol. 17 (2014), #14.2.7.
E. Biondi, L. Divieti, and G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math., Vol. 22, No. 1 (1970), pp. 22-35.
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 97.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 641.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 101.
Helen K. Jenne, Proofs you can count on, Honors Thesis, Math. Dept., Whitman College, 2013.
B. D. Josephson and J. M. Boardman, Problems Drive 1961, Eureka, The Journal of the Archimedeans, Vol. 24 (1961), p. 20; entire volume.
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Eric Weisstein's World of Mathematics, Uniform Sum Distribution.
Index entries for sequences related to factorial base representation
Index entries for sequences related to factorial numbers

Apart from initial terms, same as A059171.
Equals the square root of the first right hand column of A162990. - Johannes W. Meijer, Jul 21 2009
From a(2)=3 onward the second topmost row of arrays A276588 and A276955.
Cf. sequences with formula (n + k)*n! listed in A282466, A334397.

[Factorial(n)+Factorial(n+1): n in [0..30]]; // Vincenzo Librandi, Aug 08 2014

Maple
```
seq(n!+(n-1)!,n=1..25);
```

Table[n! + (n + 1)!, {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Apr 14 2011 *)
Total/@Partition[Range[0, 20]!, 2, 1] (* Harvey P. Dale, Nov 29 2013 *)

a(n)=denominator(polcoeff((x-1)*exp(x+x*O(x^(n+1))),n+1)); \\ Gerry Martens, Aug 12 2015

vector(30, n, (n+1)*(n-1)!) \\ Michel Marcus, Aug 12 2015

a(n) = (n+1)*(n-1)!.
E.g.f.: x/(1-x) - log(1-x). - Ralf Stephan, Apr 11 2004
The sequence 1, 3, 8, ... has g.f. (1+x-x^2)/(1-x)^2 and a(n) = n!(n + 2 - 0^n) = n!A065475(n) (offset 0). - Paul Barry, May 14 2004
a(n) = (n+1)!/n. - Claude Lenormand (claude.lenormand(AT)free.fr), Aug 24 2003
Factorial expansion of 1: 1 = sum_{n > 0} 1/a(n) [Jolley eq 302]. - Claude Lenormand (claude.lenormand(AT)free.fr), Aug 24 2003
a(1) = 2, a(2) = 3, D-finite recurrence a(n) = (n^2 - n - 2)*a(n-2) for n >= 3. - Jaume Oliver Lafont, Dec 01 2009
a(n) = ((n+2)A052649(n) - A052649(n+1))/2. - Gary Detlefs, Dec 16 2009
G.f.: U(0) where U(k) = 1 + (k+1)/(1 - x/(x + 1/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 25 2012
G.f.: 2*(1+x)/x/G(0) - 1/x, where G(k)= 1 + 1/(1 - x*(2*k+2)/(x*(2*k+2) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
a(n) = (n-1)*a(n-1) + (n-1)!. - Bruno Berselli, Feb 22 2017
a(1)=2, a(2)=3, D-finite recurrence a(n) = (n-1)*a(n-1) + (n-2)*a(n-2). - Dale Gerdemann, Jul 26 2019
a(n) = 2*A000255(n-1) + A096654(n-2). - Dale Gerdemann, Jul 26 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - 2/e (A334397). - Amiram Eldar, Jan 13 2021

More terms from James Sellers, Sep 19 2000

A001819 Central factorial numbers: second right-hand column of triangle A008955.

Original entry on oeis.org

0, 1, 5, 49, 820, 21076, 773136, 38402064, 2483133696, 202759531776, 20407635072000, 2482492033152000, 359072203696128000, 60912644957448192000, 11977654199703478272000, 2702572249389834608640000

Offset: 0

N. J. A. Sloane

Coefficient of x^2 in Product_{k=0..n}(x + k^2). - Ralf Stephan, Aug 22 2004
p divides a(p-1) for prime p > 3. p divides a((p-1)/2) for prime p > 3. For prime p, p^2 divides a(n) for n > 2*p+1. - Alexander Adamchuk, Jul 11 2006; last comment corrected by Michel Marcus, May 20 2020
The ratio a(n)/A001044(n) is the partial sum of the reciprocals of squares. E.g., a(4)/A001044(4) = 820/576 = 1/1 + 1/4 + 1/9 + 1/16. - Pierre CAMI, Oct 30 2006
a(n) is the (n-1)-st elementary symmetric function of the squares of the first n numbers. - Anton Zakharov, Nov 06 2016
Primes p such that p^2 | a(p-1) are the Wolstenholme primes A088164. - Amiram Eldar and Thomas Ordowski, Aug 08 2019

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..253 (terms 0..50 from T. D. Noe)
Takao Komatsu, Convolution identities of poly-Cauchy numbers with level 2, arXiv:2003.12926 [math.NT], 2020.
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
Index entries for sequences related to factorial numbers

Cf. A000254, A001044, A002455, A007406, A007407, A066989, A024167, A142995.
Second right-hand column of triangle A008955.
Equals row sums of A162990(n)/(n+1)^2 for n >= 1.

Table[Sum[1/i^2,{i,1,n}]/Product[1/i^2,{i,1,n}],{n,1,40}] (* Alexander Adamchuk, Jul 11 2006 *)
Table[n!^2*HarmonicNumber[n, 2], {n, 0, 15}] (* Jean-François Alcover, May 09 2012, after Joe Keane *)

a(n)=n!^2*sum(k=1,n,1/k^2) \\ Charles R Greathouse IV, Nov 06 2016

a_n = (n!)^2 * Sum_{k=1..n} 1/k^2. - Joe Keane (jgk(AT)jgk.org)
a(n) ~ (1/3)*Pi^3*n*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
Sum_{n>=0} a(n)*x^n/n!^2 = polylog(2, x)/(1-x). - Vladeta Jovovic, Jan 23 2003
a(n) = Sum_{i=1..n} 1/i^2 / Product_{i=1..n} 1/i^2. - Alexander Adamchuk, Jul 11 2006
a(0) = 0, a(n) = a(n-1)*n^2 + A001044(n-1). E.g., a(1) = 0*1 + 1 = 1 since A001044(0) = 1; a(2) = 1*2^2 + 1 = 5 since A001044(1) = 1; a(3) = 5*3^2 + 4 = 49 since A001044(2) = 4; and so on. - Pierre CAMI, Oct 30 2006
Recurrence: a(0) = 0, a(1) = 1, a(n+1) = (2*n^2 + 2*n + 1)*a(n) - n^4*a(n-1). The sequence b(n) = n!^2 satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 1. Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(1 - 1^4/(5 - 2^4/(13 - 3^4/(25 - ... -(n-1)^4/((2*n^2 - 2*n + 1)))))), leading to the infinite continued fraction expansion zeta(2) = 1/(1-1^4/(5 - 2^4/(13 - 3^4/(25 - ... - n^4/((2*n^2 + 2*n + 1) - ...))))). Compare with A142995. Compare also with A024167 and A066989. - Peter Bala, Jul 18 2008
a(n)/(n!)^2 -> zeta(2) = A013661 as n -> infinity, rewriting the Keane formula. - Najam Haq (njmalhq(AT)yahoo.com), Jan 13 2010
a(n) = s(n+1,2)^2 - 2*s(n+1,1)*s(n+1,3), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 03 2012

Minor edits by Vaclav Kotesovec, Jan 28 2015

A035287 Number of ways to place a non-attacking white and black rook on n X n chessboard.

Original entry on oeis.org

0, 4, 36, 144, 400, 900, 1764, 3136, 5184, 8100, 12100, 17424, 24336, 33124, 44100, 57600, 73984, 93636, 116964, 144400, 176400, 213444, 256036, 304704, 360000, 422500, 492804, 571536, 659344, 756900, 864900, 984064, 1115136, 1258884

Offset: 1

Erich Friedman

a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n} such that for fixed different x_1, x_2 in {1,2,3,4} and fixed y_1, y_2 in {1,2,...,n} we have f(x_1) <> y_1 and f(x_2) <> y_2. - Milan Janjic, Apr 17 2007
The third differences of certain values of the hypergeometric function 3F2 lead to this sequence, i.e., 3F2([1,n+1,n+1], [n+2,n+2], z=1) - 3*3F2([1,n+2,n+2], [n+3,n+3], z=1) + 3*3F2([1,n+3,n+3], [n+4,n+4], z=1) - 3F2([1,n+4,n+4], [n+5,n+5], z=1) = (1/((n+2)*(n+3)))^2 with n = -1, 0, 1, 2, ... See also A162990. - Johannes W. Meijer, Jul 21 2009
a(n) is the denominator (m*n)^2 of the term (1/m^2 - 1/n^2) = (2*n-1)/(m*n)^2, n = m+1, m > 0 in the Rydberg formula, while A005408 is the numerator 2n-1. So the quotient A005408/A035287 simulates the hydrogen spectral series of all hydrogen-like elements. - Freimut Marschner, Aug 10 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Leo Tavares, Illustration: Square of squares
Wolfram Research, Hypergeometric Function 3F2, The Wolfram Functions site. [From _Johannes W. Meijer_, Jul 21 2009]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002378.

Cf. A000290.

[n^2 * (n-1)^2: n in [1..40]]; // Vincenzo Librandi, May 21 2011

Table[(n - 1)^2 n^2, {n, 30}] (* Alonso del Arte, May 20 2011 *)

for n in range(100):
  print(((n+1)*n)**2) # John H. Chakkour, Dec 14 2019

[n^2*(n-1)^2 for n in range(1, 35)] # Zerinvary Lajos, Dec 03 2009

a(n) = n^2 * (n-1)^2.
a(n) = A002378(n-1)^2. - Zerinvary Lajos, Apr 11 2006
From Stephen Crowley, Jul 19 2009: (Start)
a(n) = n!*(2*n+1) / lim_{x->0} (d^n/dx^n) (polylog(2,x)*(1-1/x));
Sum_{n >= 2} 1/a(n) = 2*zeta(2) - 3 = A145426. (End) [Comment from Jianing Song, Dec 31 2022: Note that polylog(2,x)*(1-1/x) = -1 + Sum_{n>=1} ((2*n+1)/(n^2*(n+1)^2))*x^n, so (d^n/dx^n) (polylog(2,x)*(1-1/x)) = n!*(2*n+1)/(n^2*(n+1)^2) for n >= 1. - Jianing Song, Dec 31 2022]
a(n) = 4*A000537(n-1) = 2*A163102(n-1). - Omar E. Pol, Nov 29 2011
G.f.: 4*x^2*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Apr 04 2012
a(n) = 4*A000217(n-1)^2. - J. M. Bergot, Nov 01 2012
E.g.f.: x^2*(2 + 4*x + x^2)*exp(x). - Ilya Gutkovskiy, May 24 2016
Sum_{n>=2} (-1)^n/a(n) = 3 - 4*log(2). - Amiram Eldar, Jul 02 2020
Product_{n>=2} (1 - 1/a(n)) = -cos(sqrt(5)*Pi/2)*cosh(sqrt(3)*Pi/2)/Pi^2. - Amiram Eldar, Jan 29 2021
(n^2)^2 + (n^2+1)^2 + ... + (n^2 + n)^2 + a(n) = (n^2 + n + 1)^2 + ... + (n^2 + 2*n)^2. - Charlie Marion, Jun 18 2022
a(n) = A000290(n-1) * A000290(n). - Leo Tavares, Dec 03 2022
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n >= 6. - Jianing Song, Dec 30 2022

A052747 a(0) = a(1) = a(2) = 0; a(n) = n!/(n-2) for n > 2.

Original entry on oeis.org

0, 0, 0, 6, 12, 40, 180, 1008, 6720, 51840, 453600, 4435200, 47900160, 566092800, 7264857600, 100590336000, 1494484992000, 23712495206400, 400148356608000, 7155594141696000, 135161222676480000, 2688996956405760000, 56200036388880384000, 1231048416137379840000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A simple grammar.

Amiram Eldar, Table of n, a(n) for n = 0..450
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 703.

Equals the square root of the second right hand column of A162990 for n=>3. [Johannes W. Meijer, Jul 21 2009]

spec := [S,{B=Cycle(Z),S=Prod(Z,Z,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

a[n_] := If[n < 3, 0, n!/(n-2)]; Array[a, 20, 0] (* Amiram Eldar, Oct 07 2020 *)

a(n)=if(n>2,n!/(n-2),0) \\ Charles R Greathouse IV, Nov 20 2011

E.g.f.: log(-1/(-1+x))*x^2
Recurrence: {a(0)=0, a(1)=0, a(2)=0, a(3)=6, (n+2-n^2)*a(n)+(n-1)*a(n+1)}
Sum_{n>=3} 1/a(n) = 3 - e. - Amiram Eldar, Oct 07 2020
Sum_{n>=3} (-1)^(n+1)/a(n) = 3/e - 1. - Amiram Eldar, Aug 20 2022

Better description from Olivier Gérard, Jun 13 2001

cp's OEIS Frontend

A162995 A scaled version of triangle A162990.

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A162993 The second left hand column of triangle A162990.

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A162991 The first right hand column of triangle A162990.

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A162992 The second right hand column of triangle A162990.

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A162994 The third left hand column of triangle A162990.

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A002378 Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).

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A001048 a(n) = n! + (n-1)!.

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