A096338 -id:A096338 - cp's OEIS frontend

A139600 Square array T(n,k) = n(k-1)k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 9, 10, 5, 0, 1, 6, 12, 16, 15, 6, 0, 1, 7, 15, 22, 25, 21, 7, 0, 1, 8, 18, 28, 35, 36, 28, 8, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 10, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11

Offset: 0

Omar E. Pol, Apr 27 2008

A general formula for polygonal numbers is P(n,k) = (n-2)*(k-1)*k/2 + k, where P(n,k) is the k-th n-gonal number.

The triangle sums, see A180662 for their definitions, link this square array read by antidiagonals with twelve different sequences, see the crossrefs. Most triangle sums are linear sums of shifted combinations of a sequence, see e.g. A189374. - Johannes W. Meijer, Apr 29 2011

			The square array of nonnegatives together with polygonal numbers begins:
=========================================================
....................... A   A   .   .   A    A    A    A
....................... 0   0   .   .   0    0    1    1
....................... 0   0   .   .   1    1    3    3
....................... 0   0   .   .   6    7    9    9
....................... 0   0   .   .   9    3    6    6
....................... 0   1   .   .   5    2    0    0
....................... 4   2   .   .   7    9    6    7
=========================================================
Nonnegatives . A001477: 0,  1,  2,  3,  4,   5,   6,   7, ...
Triangulars .. A000217: 0,  1,  3,  6, 10,  15,  21,  28, ...
Squares ...... A000290: 0,  1,  4,  9, 16,  25,  36,  49, ...
Pentagonals .. A000326: 0,  1,  5, 12, 22,  35,  51,  70, ...
Hexagonals ... A000384: 0,  1,  6, 15, 28,  45,  66,  91, ...
Heptagonals .. A000566: 0,  1,  7, 18, 34,  55,  81, 112, ...
Octagonals ... A000567: 0,  1,  8, 21, 40,  65,  96, 133, ...
9-gonals ..... A001106: 0,  1,  9, 24, 46,  75, 111, 154, ...
10-gonals .... A001107: 0,  1, 10, 27, 52,  85, 126, 175, ...
11-gonals .... A051682: 0,  1, 11, 30, 58,  95, 141, 196, ...
12-gonals .... A051624: 0,  1, 12, 33, 64, 105, 156, 217, ...
...
=========================================================
The column with the numbers 2, 3, 4, 5, 6, ... is formed by the numbers > 1 of A000027. The column with the numbers 3, 6, 9, 12, 15, ... is formed by the positive members of A008585.

Alois P. Heinz, Rows n = 0..140, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.
Index to sequences related to polygonal numbers

Cf. A001477, A057145, A139601, A139617, A139618, A139620, A180662.
A formal extension negative n is in A326728.
Triangle sums (see the comments): A055795 (Row1), A080956 (Row2; terms doubled), A096338 (Kn11, Kn12, Kn13, Fi1, Ze1), A002624 (Kn21, Kn22, Kn23, Fi2, Ze2), A000332 (Kn3, Ca3, Gi3), A134393 (Kn4), A189374 (Ca1, Ze3), A011779 (Ca2, Ze4), A101357 (Ca4), A189375 (Gi1), A189376 (Gi2), A006484 (Gi4). - Johannes W. Meijer, Apr 29 2011
Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).

T:= func< n,k | k*(n*(k-1)+2)/2 >;
A139600:= func< n,k | T(n-k, k) >;
[A139600(n,k): k in  [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024

T:= (n, k)-> n*(k-1)*k/2+k:
seq(seq(T(d-k, k), k=0..d), d=0..14);  # Alois P. Heinz, Oct 14 2018

T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[T[n - k - 1, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)

def A139600Row(n):
    x, y = 1, 1
    yield 0
    while True:
        yield x
        x, y = x + y + n, y + n
for n in range(8):
    R = A139600Row(n)
    print([next(R) for  in range(11)]) # _Peter Luschny, Aug 04 2019

def T(n,k): return k*(n*(k-1)+2)/2
def A139600(n,k): return T(n-k, k)
flatten([[A139600(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

T(n,k) = n*(k-1)*k/2+k.
T(n,k) = A057145(n+2,k). - R. J. Mathar, Jul 28 2016
From Stefano Spezia, Apr 12 2024: (Start)
G.f.: y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3).
E.g.f.: exp(x+y)*y*(2 + x*y)/2. (End)

Edited by Omar E. Pol, Jan 05 2009

A058187 Expansion of (1+x)/(1-x^2)^4: duplicated tetrahedral numbers.

Original entry on oeis.org

1, 1, 4, 4, 10, 10, 20, 20, 35, 35, 56, 56, 84, 84, 120, 120, 165, 165, 220, 220, 286, 286, 364, 364, 455, 455, 560, 560, 680, 680, 816, 816, 969, 969, 1140, 1140, 1330, 1330, 1540, 1540, 1771, 1771, 2024, 2024, 2300, 2300, 2600, 2600, 2925, 2925, 3276, 3276

Offset: 0

Henry Bottomley, Nov 20 2000

For n >= i, i = 6,7, a(n - i) is the number of incongruent two-color bracelets of n beads, i of which are black (cf. A005513, A032280), having a diameter of symmetry. The latter means the following: if we imagine (0,1)-beads as points (with the corresponding labels) dividing a circumference of a bracelet into n identical parts, then a diameter of symmetry is a diameter (connecting two beads or not) such that a 180-degree turn of one of two sets of points around it (obtained by splitting the circumference by this diameter) leads to the coincidence of the two sets (including their labels). - Vladimir Shevelev, May 03 2011
From Johannes W. Meijer, May 20 2011: (Start)
The Kn11, Kn12, Kn13, Fi1 and Ze1 triangle sums, see A180662 for their definitions, of the Connell-Pol triangle A159797 are linear sums of shifted versions of the duplicated tetrahedral numbers, e.g., Fi1(n) = a(n-1) + 5*a(n-2) + a(n-3) + 5*a(n-4).
The Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Fi1, Fi2, Ze1 and Ze2 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above. (End)
The number of quadruples of integers [x, u, v, w] that satisfy x > u > v > w >= 0, n + 5 = x + u. - Michael Somos, Feb 09 2015
Also, this sequence is the fourth column in the triangle of the coefficients of the sum of two consecutive Fibonacci polynomials F(n+1, x) and F(n, x) (n>=0) in ascending powers of x. - Mohammad K. Azarian, Jul 18 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..880
Hansraj Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., Vol. 10, No. 8 (1979), 964-999.
Vladimir Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A057884. Sum of 2 consecutive terms gives A006918, whose sum of 2 consecutive terms gives A002623, whose sum of 2 consecutive terms gives A000292, which is this sequence without the duplication. Continuing to sum 2 consecutive terms gives A000330, A005900, A001845, A008412 successively.
Cf. A096338, A005513, A032280
Cf. A000292, A190717, A190718. - Johannes W. Meijer, May 20 2011

a058187 n = a058187_list !! n
a058187_list = 1 : f 1 1 [1] where
   f x y zs = z : f (x + y) (1 - y) (z:zs) where
     z = sum $ zipWith (*) [1..x] [x,x-1..1]
-- Reinhard Zumkeller, Dec 21 2011

A058187:= proc(n) option remember; A058187(n):= binomial(floor(n/2)+3, 3) end: seq(A058187(n), n=0..51); # Johannes W. Meijer, May 20 2011

a[n_]:= Length @ FindInstance[{x>u, u>v, v>w, w>=0, x+u==n+5}, {x, u, v, w}, Integers, 10^9]; (* Michael Somos, Feb 09 2015 *)
With[{tetra=Binomial[Range[30]+2,3]},Riffle[tetra,tetra]] (* Harvey P. Dale, Mar 22 2015 *)

{a(n) = binomial(n\2+3, 3)}; /* Michael Somos, Jun 07 2005 */

[binomial((n//2)+3, 3) for n in (0..60)] # G. C. Greubel, Feb 18 2022

a(n) = A006918(n+1) - a(n-1).
a(2*n) = a(2*n+1) = A000292(n) = (n+1)*(n+2)*(n+3)/6.
a(n) = (2*n^3 + 21*n^2 + 67*n + 63)/96 + (n^2 + 7*n + 11)(-1)^n/32. - Paul Barry, Aug 19 2003
a(n) = A108299(n-3,n)*(-1)^floor(n/2) for n > 2. - Reinhard Zumkeller, Jun 01 2005
Euler transform of finite sequence [1, 3]. - Michael Somos, Jun 07 2005
G.f.: 1 / ((1 - x) * (1 - x^2)^3) = 1 / ((1 + x)^3 * (1 - x)^4). a(n) = -a(-7-n) for all n in Z.
a(n) = binomial(floor(n/2) + 3, 3). - Vladimir Shevelev, May 03 2011
a(-n) = -a(n-7); a(n) = A000292(A008619(n)). - Guenther Schrack, Sep 13 2018
Sum_{n>=0} 1/a(n) = 3. - Amiram Eldar, Aug 18 2022

A034827 a(n) = 2*binomial(n,4).

Original entry on oeis.org

0, 0, 0, 0, 2, 10, 30, 70, 140, 252, 420, 660, 990, 1430, 2002, 2730, 3640, 4760, 6120, 7752, 9690, 11970, 14630, 17710, 21252, 25300, 29900, 35100, 40950, 47502, 54810, 62930, 71920, 81840, 92752, 104720, 117810, 132090, 147630, 164502, 182780

Offset: 0

N. J. A. Sloane

Also number of ways to insert two pairs of parentheses into a string of n-4 letters (allowing empty pairs of parentheses). E.g., there are 30 ways for 2 letters. Cf. A002415.
2,10,30,70, ... gives orchard crossing number of complete graph K_n. - Ralf Stephan, Mar 28 2003
If Y is a 2-subset of an n-set X then, for n>=4, a(n-1) is the number of 4-subsets and 5-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
Middle column of table on p. 6 of Feder and Garber. - Jonathan Vos Post, Apr 23 2009
Number of pairs of non-intersecting lines when each of n points around a circle is joined to every other point by straight lines. A pair of lines is considered non-intersecting if the lines do not intersect in either the interior or the boundary of a circle. - Melvin Peralta, Feb 05 2016
From a(2), convolution of the oblong numbers (A002378) with the nonnegative numbers (A001477). - Bruno Berselli, Oct 24 2016
Also the number of 3-cycles in the n-triangular honeycomb bishop graph. - Eric W. Weisstein, Aug 10 2017

Charles Jordan, Calculus of Finite Differences, Chelsea, 1965, p. 449.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Bruno Berselli, Table of n, a(n) for n = 0..1000
M. Aganagic, A. Klemm and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web, arXiv:hep-th/0105045, 2001.
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
Elie Feder and David Garber, The Orchard crossing number of an abstract graph, arXiv:math/0303317 [math.CO], 2003-2009.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), pp. 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

A diagonal of A088617.
Cf. A033487, A050534, A060008.
Partial sums of A007290.
Cf. A001477, A002378.
Cf. A051843 (4-cycles in the triangular honeycomb bishop graph), A290775 (5-cycles), A290779 (6-cycles).

[2*Binomial(n,4): n in [0..40]]; // Vincenzo Librandi, Oct 20 2013

[seq(binomial(n,4)*2,n=0..40)]; # Zerinvary Lajos, Jul 18 2006

CoefficientList[Series[2 x^4/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 20 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 0, 0, 2}, 50] (* Harvey P. Dale, Jun 09 2016 *)
Table[2 Binomial[n, 4], {n, 0, 40}] (* Bruno Berselli, Oct 24 2016 *)
2 Binomial[Range[0, 20], 4] (* Eric W. Weisstein, Aug 10 2017 *)

a(n)=2*binomial(n,4) \\ Charles R Greathouse IV, Jun 23 2015

a(n) = A096338(2*n-6) = 2*A000332(n), n>2. - R. J. Mathar, Nov 08 2010
G.f.: 2*x^4/(1-x)^5. - Colin Barker, Feb 29 2012
a(n) = Sum_{k=1..n-3} ( Sum_{i=1..k} i*(2*k-n+4) ). - Wesley Ivan Hurt, Sep 26 2013
E.g.f.: x^4*exp(x)/12. - G. C. Greubel, Feb 23 2017
From Amiram Eldar, Jul 19 2022: (Start)
Sum_{n>=4} 1/a(n) = 2/3.
Sum_{n>=4} (-1)^n/a(n) = 16*log(2) - 32/3. (End)

A028723 a(n) = (1/4)floor(n/2)floor((n-1)/2)floor((n-2)/2)floor((n-3)/2).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, 100, 150, 225, 315, 441, 588, 784, 1008, 1296, 1620, 2025, 2475, 3025, 3630, 4356, 5148, 6084, 7098, 8281, 9555, 11025, 12600, 14400, 16320, 18496, 20808, 23409, 26163, 29241, 32490, 36100, 39900, 44100, 48510, 53361, 58443

Offset: 0

N. J. A. Sloane

It is not known whether A000241 and this sequence agree.
Conjectured to be crossing number of complete graph K_n, see A000241.
a(n+1) is the maximum number of rectangles that can be formed from n lines. - Erich Friedman
Number of symmetric Dyck paths of semilength n and having five peaks. E.g., a(6)=3 because we have U*DU*DUU*DDU*DU*D, U*DUU*DU*DU*DDU*D and UU*DU*DU*DU*DU*DD, where U=(1,1), D=(1,-1) and * indicates a peak. - Emeric Deutsch, Jan 12 2004
a(n-5) is the number of length n words, w(1), w(2), ..., w(n) on alphabet {0,1,2} such that w(i) >= w(i+2) for all i. - Geoffrey Critzer, Mar 15 2014
a(n-1) is the number of length n binary strings beginning with a 1 that have exactly two pairs of consecutive 0's and two pairs of consecutive 1's. - Jeremy Dover, Jul 04 2016
Consider the partitions of n into two parts (p,q). Then 2*a(n+2) represents the total volume of all rectangular prisms with dimensions p, q and |q - p|. - Wesley Ivan Hurt, Apr 12 2018
a(n+1) is the number of subsets of {1, 2, ..., n} that contain 2 odd and 2 even numbers.  For example, for n = 6, a(7) = 9 and the 9 subsets are {1,2,3,4}, {1,2,3,6}, {1,2,4,5}, {1,2,5,6}, {1,3,4,6}, {1,4,5,6}, {2,3,4,5}, {2,3,5,6}, {3,4,5,6}. - Enrique Navarrete, Dec 22 2019
a(n+1) is the maximum number of induced 4-cycles in an n-node graph (Pippenger and Golumbic 1975). - Pontus von Brömssen, Mar 27 2022

			G.f. = x^5 + 3*x^6 + 9*x^7 + 18*x^8 + 36*x^9 + 60*x^10 + 100*x^11 + ...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.18, p. 533.
Martin Gardner, Knotted Doughnuts and Other Mathematical Entertainments, W. H. Freeman & Company, 1986, Chapter 11, pages 133-144.
Carsten Thomassen, Embeddings and Minors, in: R. L. Graham, M. Grötschel, and L. Lovász, Handbook of Combinatorics, Vol. 1, Elsevier, 1995, p. 314.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Bernardo M. Abrego, Oswin Aichholzer, Silvia Fernandez-Merchant, Pedro Ramos, and Gelasio Salazar, The 2-Page Crossing Number of K_n, arXiv:1206.5669 [math.CO], 2012.
Bernardo M. Abrego, Oswin Aichholzer, Silvia Fernandez-Merchant, Pedro Ramos, and Gelasio Salazar, The 2-Page Crossing Number of K_n, Discrete Comput. Geom., Vol. 49, No. 4 (2013), pp. 747-777. MR3068573.
James Dolan et al., su(3) and Zarankiewicz's conjecture.
Dhruv Mubayi, Counting substructures II: Hypergraphs, Combinatorica, Vol. 33, No. 5 (2013), pp. 591--612. MR3132928
Nicholas Pippenger and Martin Charles Golumbic, The inducibility of graphs, Journal of Combinatorial Theory Series B 19 (1975), 189-203.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A000241, A006918.

[(n^4-8*n^3+18*n^2-12*n+2*n*(n-2)*((1+(-1)^n)/2)+(2*n-3)^2*((1-(-1)^n)/2))/64: n in [0..50]]; // Vincenzo Librandi, Mar 23 2014

A028723:=n->(1/4)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2)*floor((n-3)/2); seq(A028723(n), n=0..100); # Wesley Ivan Hurt, Nov 01 2013

Table[If[EvenQ[n], n(n-2)^2(n-4)/64, (n-1)^2(n-3)^2/64], {n, 0, 50}]
Table[(n^4 -8n^3 +18n^2 -12n + 2n(n-2)((1+(-1)^n)/2) +(2n-3)^2((1-(-1)^n)/2))/64, {n, 0, 50}] (* Vincenzo Librandi, Mar 23 2014 *)
LinearRecurrence[{2, 2,-6,0,6,-2,-2,1}, {0,0,0,0,0,1,3,9}, 50] (* Harvey P. Dale, Sep 13 2018 *)
Times@@@Table[Floor[(n-k)/2], {n,0,60}, {k,0,3}]/4 (* Eric W. Weisstein, Apr 29 2019 *)

a(n) = if (n % 2, (n-1)^2 *(n-3)^2/64, n*(n-2)^2 *(n-4)/64); \\ Michel Marcus, Nov 02 2013

{a(n) = prod(k=0, 3, (n - k) \ 2) / 4}; /* Michael Somos, Nov 02 2014 */

[(n*(-12 +18*n -8*n^2 +n^3) +2*n*(n-2)*((n+1)%2) +(2*n-3)^2*(n%2))/64 for n in (0..60)] # G. C. Greubel, Apr 08 2022

If n even, n*(n-2)^2*(n-4)/64; if n odd, (n-1)^2*(n-3)^2/64.
G.f.: x^5*(1+x+x^2)/((1-x)^5*(1+x)^3). - Emeric Deutsch, Jan 12 2004
For n>2, a(n) = A007590(n-3)*A007590(n-1)/16. - Richard R. Forberg, Dec 03 2013
a(n) = (n^4 -8*n^3 +18*n^2 -12*n +2*n*(n-2)*((1+(-1)^n)/2) + (2*n-3)^2*((1-(-1)^n)/2))/64. - Luce ETIENNE, Mar 22 2014
Euler transform of length 3 sequence [3, 3, -1]. - Michael Somos, Nov 02 2014
a(n) = a(4-n) for all n in Z. - Michael Somos, Nov 02 2014
0 = -3 + a(n) - a(n+1) - 3*a(n+2) + 3*a(n+3) + 3*a(n+4) - 3*a(n+5) - a(n+6) + a(n+7) for all n in Z. - Michael Somos, Nov 02 2014
0 = a(n)*(+a(n+2) + a(n+3)) + a(n+1)*(-3*a(n+2) +a(n+3)) for all n in Z. - Michael Somos, Nov 02 2014
a(n+1)^2 - a(n)*a(n+2) = binomial(n/2, 2)^3 for all even n in Z ( = 0 if n odd). - Michael Somos, Nov 02 2014
a(n)*(a(n+1) + a(n+2)) +a(n+1)*(-3*a(n+1) + a(n+2)) = 0 for all even n in Z ( = k^4 * (k^2 - 1) / 4 if n = 2*k + 1). - Michael Somos, Nov 02 2014
a(n) = binomial(n/2,2)^2, n even; a(n) = binomial((n-1)/2,2)*binomial((n+1)/2,2), n odd. - Enrique Navarrete, Dec 22 2019
E.g.f.: (1/128)*exp(-x)*(exp(2*x)*(9 - 12*x + 8*x^2 - 4*x^3 + 2*x^4) - 9 - 6*x - 2*x^2). - Stefano Spezia, Dec 27 2019
a(n) = A002620(n-1)*A002620(n-3)/4. - R. J. Mathar, Mar 23 2021
a(n)= A096338(n-6)+A096338(n-5)+A096338(n-4). - R. J. Mathar, Mar 23 2021
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=5} 1/a(n) = 2*Pi^2/3 - 5.
Sum_{n>=5} (-1)^(n+1)/a(n) = 2*Pi^2 - 19. (End)

A038163 G.f.: 1/((1-x)*(1-x^2))^3.

Original entry on oeis.org

1, 3, 9, 19, 39, 69, 119, 189, 294, 434, 630, 882, 1218, 1638, 2178, 2838, 3663, 4653, 5863, 7293, 9009, 11011, 13377, 16107, 19292, 22932, 27132, 31892, 37332, 43452, 50388, 58140, 66861, 76551, 87381, 99351, 112651, 127281

Offset: 0

N. J. A. Sloane

Number of symmetric nonnegative integer 6 X 6 matrices with sum of elements equal to 4*n, under action of dihedral group D_4. - Vladeta Jovovic, May 14 2000
Equals the triangular sequence convolved with the aerated triangular sequence, [1, 0, 3, 0, 6, 0, 10, ...]. - Gary W. Adamson, Jun 11 2009
Number of partitions of n (n>=1) into 1s and 2s if there are three kinds of 1s and three kinds of 2s. Example: a(2)=9 because we have 11, 11', 11", 1'1', 1'1", 1"1", 2, 2', and 2". - Emeric Deutsch, Jun 26 2009
Equals the tetrahedral numbers with repeats convolved with the natural numbers: (1 + x + 4x^2 + 4x^3 + ...) * (1 + 2x + 3x^2 + 4x^3 + ...) = (1 + 3x + 9x^2 + 19x^3 + ...). - Gary W. Adamson, Dec 22 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-8,6,6,-8,0,3,-1).

Cf. A008619, A006918, A001753.
Cf. A096338.
Column k=3 of A210391. - Alois P. Heinz, Mar 22 2012
Cf. A000217.

import Data.List (inits, intersperse)
a038163 n = a038163_list !! n
a038163_list = map
    (sum . zipWith (*) (intersperse 0 $ tail a000217_list) . reverse) $
    tail $ inits $ tail a000217_list where
-- Reinhard Zumkeller, Feb 27 2015

G := 1/((1-x)^3*(1-x^2)^3): Gser := series(G, x = 0, 42): seq(coeff(Gser, x, n), n = 0 .. 37); # Emeric Deutsch, Jun 26 2009
# alternative
A038163 := proc(n)
    (4*n^5+90*n^4+760*n^3+2970*n^2+5266*n+3285+(-1)^n*(30*n^2+270*n+555))/3840 ;
end proc:
seq(A038163(n),n=0..30) ; # R. J. Mathar, Feb 22 2021

CoefficientList[Series[1/((1-x)*(1-x^2))^3, {x, 0, 40}], x] (* Jean-François Alcover, Mar 11 2014 *)
LinearRecurrence[{3,0,-8,6,6,-8,0,3,-1},{1,3,9,19,39,69,119,189,294},50] (* Harvey P. Dale, Nov 24 2022 *)

a(2*k) = (4*k + 5)*binomial(k + 4, 4)/5 = A034263(k); a(2*k + 1) = binomial(k + 4, 4)*(15 + 4*k)/5 = A059599(k), k >= 0.
a(n) = (1/3840)*(4*n^5 + 90*n^4 + 760*n^3 + 2970*n^2 + 5266*n + 3285 + (-1)^n*(30*n^2 + 270*n + 555)). Recurrence: a(n) = 3*a(n-1) - 8*a(n-3) + 6*a(n-4) + 6*a(n-5) - 8*a(n-6) + 3*a(n-8) - a(n-9). - Vladeta Jovovic, Apr 24 2002
a(n+1) - a(n) = A096338(n+2). - R. J. Mathar, Nov 04 2008

A060099 G.f.: 1/((1-x^2)^3*(1-x)^4).

Original entry on oeis.org

1, 4, 13, 32, 71, 140, 259, 448, 742, 1176, 1806, 2688, 3906, 5544, 7722, 10560, 14223, 18876, 24739, 32032, 41041, 52052, 65429, 81536, 100828, 123760, 150892, 182784, 220116, 263568, 313956, 372096, 438957, 515508, 602889, 702240, 814891, 942172, 1085623

Offset: 0

Wolfdieter Lang, Apr 06 2001

Fourth column (m=3) of triangle A060098.
Partial sums of A038163.
Equals the tetrahedral numbers, [1, 4, 10, 20, ...] convolved with the aerated triangular numbers, [1, 0, 3, 0, 6, 0, 10, ...]. [Gary W. Adamson, Jun 11 2009]

B. Broer, Hilbert series for modules of covariants, in Algebraic Groups and Their Generalizations..., Proc. Sympos. Pure Math., 56 (1994), Part I, 321-331. See p. 329.

Peter J. C. Moses, Table of n, a(n) for n = 0..9999
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 20.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-8,14,0,-14,8,3,-4,1).

Cf. A002620, A002624, A096338.
Cf. A001752 (for the similar series 1/((1-x)^4*(1-x^2))).
Cf. A028346 (for the similar series 1/((1-x)^4*(1-x^2)^2)).

a[n_]:=If[OddQ[n],((1+n) (3+n) (5+n)^2 (7+n) (9+n))/5760,((2+n) (4+n) (6+n) (8+n) (15+10 n+n^2))/5760]; Map[a,Range[0,100]] (* Peter J. C. Moses, Mar 24 2013 *)
CoefficientList[Series[1/((1-x^2)^3*(1-x)^4),{x,0,100}],x] (* Peter J. C. Moses, Mar 24 2013 *)
LinearRecurrence[{4,-3,-8,14,0,-14,8,3,-4,1},{1,4,13,32,71,140,259,448,742,1176},40] (* Harvey P. Dale, Apr 06 2018 *)

a(n) = Sum_{} A060098(n+3, 3).

G.f.: 1/((1-x)^7*(1+x)^3).

A105438 Triangle, row sums = (Fibonacci numbers - 2).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 6, 5, 2, 1, 6, 9, 8, 6, 2, 1, 7, 12, 14, 10, 7, 2, 1, 8, 16, 20, 20, 12, 8, 2, 1, 9, 20, 30, 30, 27, 14, 9, 2, 1, 10, 25, 40, 50, 42, 35, 16, 10, 2, 1, 11, 30, 55, 70, 77, 56, 44, 18, 11, 2, 1

Offset: 0

Gary W. Adamson, Apr 09 2005

Row sums = 1, 3, 6, 11, 19, 32, 53...(Fibonacci numbers - 2; starting with F(4)) The first few rows of the triangle are:
Row sums = (Fibonacci numbers - 2; starting 1, 3, 6...).
Column 1 = A002620; Column 2 = A006918; Column 3 = A096338.
Inverse array is A105522. - Paul Barry, Apr 11 2005
Diagonal sums are A027383(n). - Philippe Deléham, Jan 16 2014

			Column 2: 1, 2, 5, 8, 14, 20, 30...is generated by using the partial sum operator on 1, 1, 3, 3, 6, 6, 10, 10...
The first few rows of the triangle are:
  1;
  2, 1;
  3, 2, 1;
  4, 4, 2, 1;
  5, 6, 5, 2, 1;
  6, 9, 8, 6, 2, 1;
  7, 12, 14, 10, 7, 2, 1;
  8, 16, 20, 20, 12, 8, 2, 1;
  9, 20, 30, 30, 27, 14, 9, 2, 1;
  10, 25, 40, 50, 42, 35, 16, 10, 2, 1;
  ...

T. Mansour and A. O. Munagi, Alternating subsets modulo m, Rocky Mt. J. Math. 42, No. 4, 1313-1325 (2012), eq. (2)

Cf. A001911, A002620, A006918, A096338, A105108.

By columns (k = 0, 1, 2...); use partial sum operator on (bin(n, k) numbers repeated).
T(n,k) = Sum_{j=0..n-k} C((j+2k)/2, k)*(1+(-1)^j)+C((j-1+2k)/2, k)*(1-(-1)^j)/2; Riordan array (1/(1-x)^2, x/(1-x^2)). - Paul Barry, Apr 11 2005
T(n,k) = T(n-1,k)+T(n-1,k-1)+T(n-2,k)-T(n-2,k-1)-T(n-3,k), T(0,0)=1, T(1,0)=2, T(1,1)= 1, T(n,k)= 0 if k<0 or if k>n. - Philippe Deléham, Jan 16 2014

A115263 Correlation triangle for floor((n+2)/2).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 3, 4, 6, 4, 3, 3, 5, 7, 7, 5, 3, 4, 6, 10, 10, 10, 6, 4, 4, 7, 11, 13, 13, 11, 7, 4, 5, 8, 14, 16, 19, 16, 14, 8, 5, 5, 9, 15, 19, 22, 22, 19, 15, 9, 5, 6, 10, 18, 22, 28, 28, 28, 22, 18, 10, 6

Offset: 0

Paul Barry, Jan 18 2006

Row sums are A096338. Diagonal sums are A115264. T(2n,n) is A005993. T(2n,n)-T(2n,n+1) is floor((n+2)/2)(1+(-1)^n)/2 (aerated n+1).

			Triangle begins
1;
1,1;
2,2,2;
2,3,3,2;
3,4,6,4,3;
3,5,7,7,5,3;

G.f.: (1+x)(1+xy)/((1-x^2)^2*(1-x^2*y^2)^2*(1-x^2*y)); Number triangle T(n, k)=sum{j=0..n, [j<=k]*floor((k-j+2)/2)*[j<=n-k]*floor((n-k-j+2)/2)}.

A157901 Triangle read by rows: A000012 * A157898.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 3, 6, 10, 8, 8, 3, 9, 16, 24, 16, 16, 4, 12, 28, 40, 56, 32, 32, 4, 16, 40, 80, 96, 128, 64, 64, 5, 20, 60, 120, 216, 224, 288, 128, 128, 5, 25, 80, 200, 336, 560, 512, 640, 256, 256, 6, 30, 110, 280, 616, 896, 1408, 1152, 1408, 512, 512

Offset: 0

Gary W. Adamson & Roger L. Bagula, Mar 08 2009

Multiplication of the lower triangular matrix A157898 from the left by A000012 means: these are partial column sums of A157898.

			First few rows of the triangle, n>=0:
  1;
  1,  1;
  2,  2,  2;
  2,  4,  4,   4;
  3,  6, 10,   8,   8;
  3,  9, 16,  24,  16,  16;
  4, 12, 28,  40,  56,  32,  32;
  4, 16, 40,  80,  96, 128,  64,  64;
  5, 20, 60, 120, 216, 224, 288, 128, 128;
  5, 25, 80, 200, 336, 560, 512, 640, 256, 256;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000012, A105635, A157898.
Columns: A004526 (k=0), A002620 (k=1), A006584 (k=2), 4*A096338 (k=3), 8*A177747 (k=4), 16*A299337 (k=5), 32*A178440 (k=6).
Sums include: A105635(n+1) (row), A166486(n+1) (alternating sign diagonal), A232801(n+1) (diagonal).

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
function t(n, k) // t = A059576
  if k eq 0 or k eq n then return A011782(n);
  else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);
  end if; return t;
end function;
A157898:= func< n, k | (&+[(-1)^(n-j)*Binomial(n, j)*t(j, k): j in [k..n]]) >;
A157071:= func< n,k | (&+[A157898(j+k,k): j in [0..n-k]]) >;
[A157071(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 27 2025

t[n_, k_]:= t[n,k]= If[k==0 || k==n, 2^(n-1) +Boole[n==0]/2, 2*t[n-1,k-1] +2*t[n-1,k] - (2 -Boole[n==2])*t[n-2,k-1]]; (* t = A059576 *)
A157898[n_, k_]:= A157898[n,k]= Sum[(-1)^(n-j)*Binomial[n,j]*t[j,k], {j,k,n}];
A157901[n_, k_]:= A157901[n,k]= Sum[A157898[j+k,k], {j,0,n-k}];
Table[A157901[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 27 2025 *)

@CachedFunction
def t(n, k): # t = A059576
    if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782
    else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)
def A157898(n, k): return sum((-1)^(n+k-j)*binomial(n, j+k)*t(j+k, k) for j in range(n-k+1))
def A157071(n,k): return sum(A157898(j+k,k) for j in range(n-k+1))
print(flatten([[A157071(n,k) for k in range(n+1)] for n in range(10)])) # G. C. Greubel, Aug 27 2025

T(n,k) = Sum_{j=0..n} A157898(j,k).

Edited by the Associate Editors of the OEIS, Apr 10 2009

More terms from G. C. Greubel, Aug 27 2025

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A139600 Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.

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A058187 Expansion of (1+x)/(1-x^2)^4: duplicated tetrahedral numbers.

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A034827 a(n) = 2*binomial(n,4).

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A028723 a(n) = (1/4)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2)*floor((n-3)/2).

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A038163 G.f.: 1/((1-x)*(1-x^2))^3.

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A060099 G.f.: 1/((1-x^2)^3*(1-x)^4).

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A139600 Square array T(n,k) = n(k-1)k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.

A028723 a(n) = (1/4)floor(n/2)floor((n-1)/2)floor((n-2)/2)floor((n-3)/2).