A055795 -id:A055795 - cp's OEIS frontend

A057145 Square array of polygonal numbers T(n,k) = ((n-2)k^2 - (n-4)k)/2, n >= 2, k >= 1, read by antidiagonals upwards.

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 9, 10, 5, 1, 6, 12, 16, 15, 6, 1, 7, 15, 22, 25, 21, 7, 1, 8, 18, 28, 35, 36, 28, 8, 1, 9, 21, 34, 45, 51, 49, 36, 9, 1, 10, 24, 40, 55, 66, 70, 64, 45, 10, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11, 1, 12, 30, 52, 75, 96, 112

Offset: 2

N. J. A. Sloane, Sep 12 2000

The set of the "nontrivial" entries T(n>=3,k>=3) is in A090466. - R. J. Mathar, Jul 28 2016

T(n,k) is the smallest number that can be expressed as the sum of k consecutive positive integers that differ by n - 2. In other words: T(n,k) is the sum of k terms of the arithmetic progression with common difference n - 2 and 1st term 1, (see the example). - Omar E. Pol, Apr 29 2020

			Array T(n k) (n >= 2, k >= 1) begins:
  1,  2,  3,  4,   5,   6,   7,   8,   9,  10,  11, ...
  1,  3,  6, 10,  15,  21,  28,  36,  45,  55,  66, ...
  1,  4,  9, 16,  25,  36,  49,  64,  81, 100, 121, ...
  1,  5, 12, 22,  35,  51,  70,  92, 117, 145, 176, ...
  1,  6, 15, 28,  45,  66,  91, 120, 153, 190, 231, ...
  1,  7, 18, 34,  55,  81, 112, 148, 189, 235, 286, ...
  1,  8, 21, 40,  65,  96, 133, 176, 225, 280, 341, ...
  1,  9, 24, 46,  75, 111, 154, 204, 261, 325, 396, ...
  1, 10, 27, 52,  85, 126, 175, 232, 297, 370, 451, ...
  1, 11, 30, 58,  95, 141, 196, 260, 333, 415, 506, ...
  1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, ...
  1, 13, 36, 70, 115, 171, 238, 316, 405, 505, 616, ...
  1, 14, 39, 76, 125, 186, 259, 344, 441, 550, 671, ...
-------------------------------------------------------
From _Wolfdieter Lang_, Nov 04 2014: (Start)
The triangle a(k, m) begins:
  k\m 1  2  3  4  5   6   7   8   9  10  11  12 13 14 ...
  2:  1
  3:  1  2
  4:  1  3  3
  5:  1  4  6  4
  6:  1  5  9 10  5
  7:  1  6 12 16 15   6
  8:  1  7 15 22 25  21   7
  9:  1  8 18 28 35  36  28   8
  10: 1  9 21 34 45  51  49  36   9
  11: 1 10 24 40 55  66  70  64  45  10
  12: 1 11 27 46 65  81  91  92  81  55  11
  13: 1 12 30 52 75  96 112 120 117 100  66  12
  14: 1 13 33 58 85 111 133 148 153 145 121  78 13
  15: 1 14 36 64 95 126 154 176 189 190 176 144 91 14
  ...
-------------------------------------------------------
a(2,1) = T(2,1), a(6, 3) = T(4, 3). (End)
.
From _Omar E. Pol_, May 03 2020: (Start)
Illustration of the corner of the square array:
.
  1       2         3           4
  O     O O     O O O     O O O O
.
  1       3         6          10
  O     O O     O O O     O O O O
          O       O O       O O O
                    O         O O
                                O
.
  1       4         9          16
  O     O O     O O O     O O O O
          O       O O       O O O
          O       O O       O O O
                    O         O O
                    O         O O
                                O
                                O
.
  1       5        12          22
  O     O O     O O O     O O O O
          O       O O       O O O
          O       O O       O O O
          O       O O       O O O
                    O         O O
                    O         O O
                    O         O O
                                O
                                O
                                O
(End)

A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 189, 1966.
J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag (Copernicus), p. 38, 1996.

T. D. Noe, Rows n = 2..100, flattened
Lukas Andritsch, Boundary algebra of a GL_m-dimer, arXiv:1804.07243 [math.RT], 2018.
Index to sequences related to polygonal numbers

Many rows and columns of this array are in the database.
Cf. A055795 (antidiagonal sums), A064808 (main diagonal).
Cf. A063212, A077028, A086270, A090466, A139600.

/* As square array: */ t:=func; [[t(s,n): s in [1..11]]: n in [2..14]]; // Bruno Berselli, Jun 24 2013

A057145 := proc(n,k)
    ((n-2)*k^2-(n-4)*k)/2 ;
end proc:
seq(seq(A057145(d-k,k),k=1..d-2),d=3..12); # R. J. Mathar, Jul 28 2016

nn = 12; Flatten[Table[k (3 - k^2 - n + k*n)/2, {n, 2, nn}, {k, n - 1}]] (* T. D. Noe, Oct 10 2012 *)

T(2n+4,n) = n^3. - Stuart M. Ellerstein (ellerstein(AT)aol.com), Aug 28 2000
T(n, k) = T(n-1, k) + k*(k-1)/2 [with T(2, k)=k] = T(n, k-1) + 1 + (n-2)*(k-1) [with T(n, 0)=0] = k + (n-2)k(k-1)/2 = k + A063212(n-2, k-1). - Henry Bottomley, Jul 11 2001
G.f. for row n: x*(1+(n-3)*x)/(1-x)^3, n>=2. - Paul Barry, Feb 21 2003
From Wolfdieter Lang, Nov 05 2014: (Start)
The triangle is a(n, m) = T(n-m+1, m) = (1/2)*m*(n*(m-1) + 3 - m^2) for n >= 2, m = 1, 2, ..., n-1 and zero elsewhere.
O.g.f. for column m (without leading zeros): (x*binomial(m,2) + (1+2*m-m^2)*(m/2)*(1-x))/(x^(m-1)*(1-x)^2). (End)
T(n,k) = A139600(n-2,k) = A086270(n-2,k). - R. J. Mathar, Jul 28 2016
Row sums of A077028: T(n+2,k+1) = Sum_{j=0..k} A077028(n,j), where A077028(n,k) = 1+n*k is the square array interpretation of A077028 (the 1D polygonal numbers). - R. J. Mathar, Jul 30 2016
G.f.: x^2*y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3). - Stefano Spezia, Apr 12 2024

a(50)=49 corrected to a(50)=40 by Jean-François Alcover, Jul 22 2011

Edited: Name shortened, offset in Paul Barry's g.f. corrected and Conway-Guy reference added. - Wolfdieter Lang, Nov 04 2014

A004006 a(n) = C(n,1) + C(n,2) + C(n,3), or n*(n^2 + 5)/6.

Original entry on oeis.org

0, 1, 3, 7, 14, 25, 41, 63, 92, 129, 175, 231, 298, 377, 469, 575, 696, 833, 987, 1159, 1350, 1561, 1793, 2047, 2324, 2625, 2951, 3303, 3682, 4089, 4525, 4991, 5488, 6017, 6579, 7175, 7806, 8473, 9177, 9919, 10700, 11521, 12383, 13287, 14234, 15225

Offset: 0

Albert D. Rich (Albert_Rich(AT)msn.com)

3-dimensional analog of centered polygonal numbers.
The Burnside group B(3,n) has order 3^a(n).
Answer to the question: if you have a tall building and 3 plates and you need to find the highest story, a plate thrown from which does not break, what is the number of stories you can handle given n tries? - Leonid Broukhis, Oct 24 2000
Equals row sums of triangle A144329 starting with "1". - Gary W. Adamson, Sep 18 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=4, a(n-1)=-coeff(charpoly(A,x),x^(n-3)). - Milan Janjic, Jan 24 2010
From J. M. Bergot, Aug 03 2011: (Start)
If one formed the 3 X 3 square
  |  n  |  n+1 | n+2 |
  | n+3 |  n+4 | n+5 |
  | n+6 |  n+7 | n+8 |
  and found the sum of the horizontal products n*(n + 1)*(n + 2) + (n + 3)*(n + 4)*(n + 5) + (n + 6)*(n + 7)*(n + 8) and added the sum of the vertical products n*(n + 3)*(n + 6) + (n + 1)*(n + 4)*(n + 7) + (n + 2)*(n + 5)(n + 8) one gets 6*n^3 + 72*n^2 + 318*n + 504. This will give 36 times the values of all the terms in this sequence. (End)
a(n) is divisible by n for n congruent to {1,5} mod 6. (see A007310). - Gary Detlefs, Dec 08 2011
From Beimar Naranjo, Feb 22 2024: (Start)
Number of compositions with at most three parts and sum at most n.
Also the number of compositions with at most one part distinct from 1 and with a sum at most n. (End)
a(n) is the number of strings of length n defined on {0, 1, 2, 3} that contain one 1 and any number of 0's, or two 2's and any number of 0's, or three 3's and any number of 0's. For example, a(6) = 41 since the strings are the 20 permutations of 333000, the 15 permutations of 220000 and the 6 permutations of 100000. - Enrique Navarrete, Jun 18 2025

			G.f. = x + 3*x^2 + 7*x^3 + 14*x^4 + 25*x^5 + 41*x^6 + 63*x^7 + 92*x^8 + ... - _Michael Somos_, Dec 29 2019

W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 1966, see p. 380.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372. [From Parthasarathy Nambi, Sep 30 2009]
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics, 2017, Vol. 41 Issue 6, pp. 849-853.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Laurent Saloff-Coste, Random walks on finite groups, in Probability on discrete structures, 263-346, Encyclopaedia Math. Sci., 110, Springer, 2004.
Bridget Eileen Tenner, Reduced word manipulation: patterns and enumeration, J. Algebr. Comb. 46, No. 1, 189-217 (2017), w in S_n(231) l(w)=3.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A051576, A055795, A006552. Differences give A000217 + 1 or A000124.
1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Cf. A000292, A001477, A007310, A128834, A144329, A228074.

List([0..50], n-> n*(n^2+5)/6); # G. C. Greubel, Aug 27 2019

a004006 n = a000292 n + n + 1  -- Reinhard Zumkeller, Mar 31 2012

[n*(n^2+5)/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

A004006 := proc(n) n*(n^2+5)/6 ;end proc:
seq(A004006(n),n=0..10) ; # R. J. Mathar, Jun 05 2011

Table[Total[Table[Binomial[n,i], {i,3}]], {n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,3,7}, 50] (* Harvey P. Dale, Aug 21 2011 *)

A004006(n):=n*(n^2+5)/6$ makelist(A004006(n),n,0,30); /* Martin Ettl, Jan 08 2013 */

{a(n) = n*(n^2 + 5)/6}; /* Michael Somos, May 04 2007 */

[n*(n^2+5)/6 for n in (0..50)] # G. C. Greubel, Aug 27 2019

G.f.: x*(1-x+x^2)/(1-x)^4.
E.g.f.: x*(1 + x/2 + x^2/6) * exp(x).
a(-n) = -a(n).
a(n) = binomial(n+2,n-1) - binomial(n,n-2). - Zerinvary Lajos, May 11 2006
Euler transform of length 6 sequence [3, 1, 1, 0, 0, -1]. - Michael Somos, May 04 2007
Starting (1, 3, 7, 14, ...) = binomial transform of [1, 2, 2, 1, 0, 0, 0, ...]. - Gary W. Adamson, Apr 24 2008
a(0)=0, a(1)=1, a(2)=3, a(3)=7, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Aug 21 2011
a(n+1) = A000292(n) + n + 1. - Reinhard Zumkeller, Mar 31 2012
a(n) = 2*a(n-1) + (n-1) - a(n-2) with a(0) = 0, a(1) = 1. - Richard R. Forberg, Jan 23 2014
a(n) = Sum_{i=1..n} binomial(n-2i,2). - Wesley Ivan Hurt, Nov 18 2017
a(n) = n + Sum_{k=0..n} k*(n-k). - Gionata Neri, May 19 2018
a(n) = Sum_{k=0..n-1} A000124(k). - Torlach Rush, Aug 05 2018
G.f.: ((1 - x^5)/(1 - x)^5 - 1)/5. - Michael Somos, Dec 29 2019
G.f.: g(f(x)), where g is g.f. of A001477 and f is g.f. of A128834. - Oboifeng Dira, Jun 21 2020
Sum_{n>0} 1/a(n) = 3*(2*gamma + polygamma(0, 1-i*sqrt(5)) + polygamma(0, 1+i*sqrt(5)))/5 = 1.6787729555834452106286261834348972248... where i denotes the imaginary unit. - Stefano Spezia, Aug 31 2023

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

Original entry on oeis.org

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

A000127 Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386, 562, 794, 1093, 1471, 1941, 2517, 3214, 4048, 5036, 6196, 7547, 9109, 10903, 12951, 15276, 17902, 20854, 24158, 27841, 31931, 36457, 41449, 46938, 52956, 59536, 66712, 74519, 82993, 92171, 102091, 112792, 124314, 136698

Offset: 1

N. J. A. Sloane

a(n) is the sum of the first five terms in the n-th row of Pascal's triangle. - Geoffrey Critzer, Jan 18 2009
{a(k): 1 <= k <= 5} = divisors of 16. - Reinhard Zumkeller, Jun 17 2009
Equals binomial transform of [1, 1, 1, 1, 1, 0, 0, 0, ...]. - Gary W. Adamson, Mar 02 2010
From Bernard Schott, Apr 05 2021: (Start)
As a(n) = 2^(n-1) for n = 1..5, it is misleading to believe that a(n) = 2^(n-1) for n > 5 (see Patrick Popescu-Pampu link); other curiosities: a(6) = 2^5 - 1 and a(10) = 2^8.
The sequence of the first differences is A000125, the sequence of the second differences is A000124, the sequence of the third differences is A000027 and the sequence of the fourth differences is the all 1's sequence A000012 (see J. H. Conway and R. K. Guy reference, p. 80). (End)
a(n) is the number of binary words of length n matching the regular expression 0*1*0*1*0*. A000124 and A000125 count binary words of the form 0*1*0* and 1*0*1*0*, respectively. - Manfred Scheucher, Jun 22 2023

			a(7)=99 because the first five terms in the 7th row of Pascal's triangle are 1 + 7 + 21 + 35 + 35 = 99. - _Geoffrey Critzer_, Jan 18 2009
G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 31*x^6 + 57*x^7 + 99*x^8 + 163*x^9 + ...

R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 28.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, Chap. 3.
J. H. Conway and R. K. Guy, Le Livre des Nombres, Eyrolles, 1998, p. 80.
J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 33 pp. 18; 128 Ellipses Paris 2004.
A. Deledicq and D. Missenard, A La Recherche des Régions Perdues, Math. & Malices, No. 22 Summer 1995 issue pp. 22-3 ACL-Editions Paris.
M. Gardner, Mathematical Circus, pp. 177; 180-1 Alfred A. Knopf NY 1979.
M. Gardner, The Colossal Book of Mathematics, 2001, p. 561.
James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
M. de Guzman, Aventures Mathématiques, Prob. B pp. 115-120 PPUR Lausanne 1990.
Ross Honsberger; Mathematical Gems I, Chap. 9.
Ross Honsberger; Mathematical Morsels, Chap. 3.
Jeux Mathématiques et Logiques, Vol. 3 pp. 12; 51 Prob. 14 FFJM-SERMAP Paris 1988.
J. N. Kapur, Reflections of a Mathematician, Chap.36, pp. 337-343, Arya Book Depot, New Delhi 1996.
C. D. Miller, V. E. Heeren, J. Hornsby, M. L. Morrow and J. Van Newenhizen, Mathematical Ideas, Tenth Edition, Pearson, Addison-Wesley, Boston, 2003, Cptr 1, 'The Art of Problem Solving, page 6.
I. Niven, Mathematics of Choice, pp. 158; 195 Prob. 40 NML 15 MAA 1965.
C. S. Ogilvy, Tomorrow's Math, pp. 144-6 OUP 1972.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 252-255.
Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Prometheus Books, NY, 2007, page 81-87.
A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.
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..1000
Alan Calvitti, Illustration of initial terms
M. L. Cornelius, Variations on a geometric progression, Mathematics in School, 4 (No. 3, May 1975), p. 32. (Annotated scanned copy)
Colin Defant, Meeting Covered Elements in nu-Tamari Lattices, arXiv:2104.03890 [math.CO], 2021.
M. Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
R. K. Guy, Letter to N. J. A. Sloane
D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.
Math Forum, Regions of a circle Cut by Chords to n points.
R. J. Mathar, The number of binary nxm matrices with at most k 1's in each row or column, (2014) Table 4 column 1.
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Leo Moser and W. Bruce Ross, Mathematical Miscellany, On the Danger of Induction, Mathematics Magazine, Vol. 23, No. 2 (Nov. - Dec., 1949), pp. 109-114.
M. Noy, A Short Solution of a Problem in Combinatorial Geometry, Mathematics Magazine, pp. 52-3 69(1) 1996 MAA.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Patrick Popescu-Pampu, Démarrage trompeur, Images des Mathématiques, CNRS, 2017, rediffusion 2021 (in French).
D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.
H. P. Robinson, Letter to N. J. A. Sloane, Mar 21 1985
Grant Sanderson, Circle division solution, video, 2015.
Grant Sanderson, Circle division solution, updated video (2023).
K. Uhland, A Blase of Glory
K. Uhland, Moser's Problem
Prasad Balakrishnan Warrier, The physiognomy of the Erdős-Szekeres conjecture (happy ending problem), Math. Student (Indian Math. Soc., 2024) Vol. 93, Nos. 3-4, 28-48.
Eric Weisstein's World of Mathematics, Circle Division by Chords
Eric Weisstein's World of Mathematics, Strong Law of Small Numbers
Reinhart Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000012, A000027, A000124, A000125, A002522, A005408, A016813, A086514, A058331, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261, A007318, A008859-A008863, A219531, A223718.

a000127 = sum . take 5 . a007318_row  -- Reinhard Zumkeller, Nov 24 2012

[(n^4-6*n^3+23*n^2-18*n+24)/24: n in [1..50]]; // Vincenzo Librandi, Feb 16 2015

A000127 := n->(n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24;
with (combstruct):ZL:=[S, {S=Sequence(U, card=1)}, unlabeled]: seq(count(subs(r=6, ZL), size=m), m=0..41); # Zerinvary Lajos, Mar 08 2008

f[n_] := Sum[Binomial[n, i], {i, 0, 4}]; Table[f@n, {n, 0, 40}] (* Robert G. Wilson v, Jun 29 2007 *)
Total/@Table[Binomial[n-1,k],{n,50},{k,0,4}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,2,4,8,16},50] (* Harvey P. Dale, Aug 24 2011 *)
Table[(n^4 - 6 n^3 + 23 n^2 - 18 n + 24) / 24, {n, 100}] (* Vincenzo Librandi, Feb 16 2015 *)
a[ n_] := Binomial[n, 4] + Binomial[n, 2] + 1; (* Michael Somos, Dec 23 2017 *)

a(n)=(n^4-6*n^3+23*n^2-18*n+24)/24 \\ Charles R Greathouse IV, Mar 22 2016

{a(n) = binomial(n, 4) + binomial(n, 2) + 1}; /* Michael Somos, Dec 23 2017 */

def A000127(n): return n*(n*(n*(n - 6) + 23) - 18)//24 + 1 # Chai Wah Wu, Sep 18 2021

a(n) = C(n-1, 4) + C(n-1, 3) + ... + C(n-1, 0) = A055795(n) + 1 = C(n, 4) + C(n-1, 2) + n.
a(n) = Sum_{k=0..2} C(n, 2k). - Joel Sanderi (sanderi(AT)itstud.chalmers.se), Sep 08 2004
a(n) = (n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24.
G.f.: (1 - 3*x + 4*x^2 - 2*x^3 + x^4)/(1-x)^5. (for offset 0) - Simon Plouffe in his 1992 dissertation
E.g.f.: (1 + x + x^2/2 + x^3/6 + x^4/24)*exp(x) (for offset 0). [Typos corrected by Juan M. Marquez, Jan 24 2011]
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 4. - Harvey P. Dale, Aug 24 2011
a(n) = A000124(A000217(n-1)) - n*A000217(n-2) - A034827(n), n > 1. - Melvin Peralta, Feb 15 2016
a(n) = A223718(-n). - Michael Somos, Dec 23 2017
For n > 2, a(n) = n + 1 + sum_{i=2..(n-2)}sum_{j=1..(n-i)}(1+(i-1)(j-1)). - Alec Jones, Nov 17 2019

Formula corrected and additional references from torsten.sillke(AT)lhsystems.com

Additional correction from Jonas Paulson (jonasso(AT)sdf.lonestar.org), Oct 30 2003

A051601 Rows of triangle formed using Pascal's rule except we begin and end the n-th row with n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 7, 8, 7, 4, 5, 11, 15, 15, 11, 5, 6, 16, 26, 30, 26, 16, 6, 7, 22, 42, 56, 56, 42, 22, 7, 8, 29, 64, 98, 112, 98, 64, 29, 8, 9, 37, 93, 162, 210, 210, 162, 93, 37, 9, 10, 46, 130, 255, 372, 420, 372, 255, 130, 46, 10

Offset: 0

Asher Auel

The number of spotlight tilings of an m X n rectangle missing the southeast corner. E.g., there are 2 spotlight tilings of a 2 X 2 square missing its southeast corner. - Bridget Tenner, Nov 10 2007
T(n,k) = A134636(n,k) - A051597(n,k). - Reinhard Zumkeller, Nov 23 2012
For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 18 2013
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013

			From _Roger L. Bagula_, Feb 17 2009: (Start)
Triangle begins:
   0;
   1,  1;
   2,  2,   2;
   3,  4,   4,   3;
   4,  7,   8,   7,    4;
   5, 11,  15,  15,   11,    5;
   6, 16,  26,  30,   26,   16,   6;
   7, 22,  42,  56,   56,   42,   22,    7;
   8, 29,  64,  98,  112,   98,   64,   29,   8;
   9, 37,  93, 162,  210,  210,  162,   93,   37,   9;
  10, 46, 130, 255,  372,  420,  372,  255,  130,  46,  10;
  11, 56, 176, 385,  627,  792,  792,  627,  385, 176,  56, 11;
  12, 67, 232, 561, 1012, 1419, 1584, 1419, 1012, 561, 232, 67, 12. ... (End)

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
B. E. Tenner, Spotlight tiling, Ann. Combinat. 14 (4) (2010) 553-568.
Index entries for triangles and arrays related to Pascal's triangle

Row sums give A000918(n+1).
Cf. A007318, A224791, A228196, A228576.
Columns from 2 to 9, respectively: A000124; A000125, A055795, A027660, A055796, A055797, A055798, A055799 (except 1 for the last seven). [Bruno Berselli, Aug 02 2013]
Cf. A001477, A162551 (central terms).

Flat(List([0..12], n-> List([0..n], k->  Binomial(n, k+1) + Binomial(n, n-k+1) ))); # G. C. Greubel, Nov 12 2019

a051601 n k = a051601_tabl !! n !! k
a051601_row n = a051601_tabl !! n
a051601_tabl = iterate
               (\row -> zipWith (+) ([1] ++ row) (row ++ [1])) [0]
-- Reinhard Zumkeller, Nov 23 2012

/* As triangle: */ [[Binomial(n,m+1)+Binomial(n,n-m+1): m in [0..n]]: n in [0..12]]; // Bruno Berselli, Aug 02 2013

seq(seq(binomial(n,k+1) + binomial(n, n-k+1), k=0..n), n=0..12); # G. C. Greubel, Nov 12 2019

T[n_, k_]:= T[n, k] = Binomial[n, k+1] + Binomial[n, n-k+1];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula, Feb 17 2009; modified by G. C. Greubel, Nov 12 2019 *)

T(n,k) = binomial(n, k+1) + binomial(n, n-k+1);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 12 2019

[[binomial(n, k+1) + binomial(n, n-k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 12 2019

T(m,n) = binomial(m+n,m) - 2*binomial(m+n-2,m-1), up to offset and transformation of array to triangular indices. - Bridget Tenner, Nov 10 2007
T(n,k) = binomial(n, k+1) + binomial(n, n-k+1). - Roger L. Bagula, Feb 17 2009
T(0,n) = T(n,0) = n, T(n,k) = T(n-1,k) + T(n-1,k-1), 0 < k < n.

A289837 Number of cliques in the n-tetrahedral graph.

Original entry on oeis.org

1, 1, 2, 16, 76, 261, 757, 2003, 5035, 12286, 29426, 69554, 162670, 376923, 865971, 1973941, 4466853, 10040524, 22430584, 49829116, 110127536, 242254321, 530619937, 1157676711, 2516640751, 5452664426, 11777687182, 25367246038, 54492508610, 116769551831

Offset: 1

Eric W. Weisstein, Jul 13 2017

Here, "cliques" means complete subgraphs (not necessarily the largest).
Sequence extended to a(1) using formula. - Andrew Howroyd, Jul 18 2017
From Gus Wiseman, Jan 11 2019: (Start)
The n-tetrahedral graph has all 3-subsets of {1,...,n} as vertices, and two are connected iff they share two elements. So a(n) is the number of 3-uniform hypergraphs on n labeled vertices where every two edges have two vertices in common. For example, the a(4) = 16 hypergraphs are:
  {}
  {{1,2,3}}
  {{1,2,4}}
  {{1,3,4}}
  {{2,3,4}}
  {{1,2,3},{1,2,4}}
  {{1,2,3},{1,3,4}}
  {{1,2,3},{2,3,4}}
  {{1,2,4},{1,3,4}}
  {{1,2,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4}}
  {{1,2,3},{1,2,4},{2,3,4}}
  {{1,2,3},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
The following are non-isomorphic representatives of the 7 unlabeled 3-uniform cliques on 6 vertices, and their multiplicities in the labeled case, which add up to a(6) = 261.
   1 X {}
  20 X {{1,2,3}}
  90 X {{1,3,4},{2,3,4}}
  60 X {{1,4,5},{2,4,5},{3,4,5}}
  60 X {{1,2,4},{1,3,4},{2,3,4}}
  15 X {{1,5,6},{2,5,6},{3,5,6},{4,5,6}}
  15 X {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Tetrahedral Graph
Index entries for linear recurrences with constant coefficients, signature (11,-52,138,-225,231,-146,52,-8).

Cf. A055795 (maximal cliques), A287232 (independent vertex sets), A290056 (triangular graph).

Cf. A000665, A125791, A299471, A302374, A302394, A322451, A323293, A323296, A323298.

Table[(2^(n - 2) - n + 1) Binomial[n, 2] + Binomial[n, 3] +
  5 Binomial[n, 4] + 1, {n, 20}] (* Eric W. Weisstein, Jul 21 2017 *)
LinearRecurrence[{11, -52, 138, -225, 231, -146, 52, -8}, {1, 1, 2, 16, 76, 261, 757, 2003}, 20] (* Eric W. Weisstein, Jul 21 2017 *)
CoefficientList[Series[(1 - 10 x + 43 x^2 - 92 x^3 + 91 x^4 - 25 x^5 - 5 x^6 - 8 x^7)/((-1 + x)^5 (-1 + 2 x)^3), {x, 0, 20}], x] (* Eric W. Weisstein, Jul 21 2017 *)
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]<=1&]],{n,6}] (* Gus Wiseman, Jan 11 2019 *)

a(n) = 1 + binomial(n,3) + (2^(n-2)-n+1)*binomial(n,2) + 5*binomial(n,4); \\ Andrew Howroyd, Jul 18 2017

Vec(x*(1 - 10*x + 43*x^2 - 92*x^3 + 91*x^4 - 25*x^5 - 5*x^6 - 8*x^7) / ((1 - x)^5*(1 - 2*x)^3) + O(x^40)) \\ Colin Barker, Jul 19 2017

a(n) = 1 + binomial(n,3) + (2^(n-2)-n+1)*binomial(n,2) + 5*binomial(n,4). - Andrew Howroyd, Jul 18 2017
a(n) = 11*a(n-1)-52*a(n-2)+138*a(n-3)-225*a(n-4)+231*a(n-5)-146*a(n-6)+52*a(n-7)-8*a(n-8). - Eric W. Weisstein, Jul 21 2017
From Colin Barker, Jul 19 2017: (Start)
G.f.: x*(1 - 10*x + 43*x^2 - 92*x^3 + 91*x^4 - 25*x^5 - 5*x^6 - 8*x^7) / ((1 - x)^5*(1 - 2*x)^3).
a(n) = (24 - (34+3*2^n)*n + (67+3*2^n)*n^2 - 38*n^3 + 5*n^4) / 24.
(End)
Binomial transform of A323294. - Gus Wiseman, Jan 11 2019

a(1)-a(5) and a(21)-a(30) from Andrew Howroyd, Jul 18 2017

A055794 Triangle T read by rows: T(i,0)=1 for i >= 0; T(i,i)=1 for i=0,1,2,3; T(i,i)=0 for i >= 4; T(i,j) = T(i-1,j) + T(i-2,j-1) for 1<=j<=i-1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 0, 1, 5, 7, 4, 1, 0, 1, 6, 11, 8, 3, 0, 0, 1, 7, 16, 15, 7, 1, 0, 0, 1, 8, 22, 26, 15, 4, 0, 0, 0, 1, 9, 29, 42, 30, 11, 1, 0, 0, 0, 1, 10, 37, 64, 56, 26, 5, 0, 0, 0, 0, 1, 11, 46, 93, 98, 56, 16, 1, 0, 0, 0, 0, 1, 12, 56, 130, 162, 112, 42, 6, 0, 0, 0, 0, 0

Offset: 0

Clark Kimberling, May 28 2000

T(i+j,j) is the number of strings (s(1),...,s(i+1)) of nonnegative integers s(k) such that 0<=s(k)-s(k-1)<=1 for k=2,3,...,i+1 and s(i+1)=j.

T(i+j,j) is the number of compositions of j consisting of i parts, all of in {0,1}.

			Triangle begins:
  1;
  1, 1;
  1, 2, 1;
  1, 3, 2, 1;
  1, 4, 4, 2, 0;
  1, 5, 7, 4, 1, 0;
  ...
T(7,4) counts the strings 3334, 3344, 3444, 2234, 2334, 2344, 1234.
T(7,4) counts the compositions 001, 010, 100, 011, 101, 110, 111.

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 1B.

Row sums: A000204 (Lucas numbers).

Cf. subsequences T(2n+m,n): A000125 (m=0, cake numbers), A055795 (m=1), A027660 (m=2), A055796 (m=3), A055797 (m=4), A055798 (m=5), A055799 (m=6).

T:= function(n,k)
    if k=0 then return 1;
    elif k=n and n<4 then return 1;
    elif k=n then return 0;
    else return T(n-1,k) + T(n-2,k-1);
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jan 25 2020

function T(n,k)
  if k eq 0 then return 1;
  elif k eq n and n lt 4 then return 1;
  elif k eq n then return 0;
  else return T(n-1,k) + T(n-2, k-1);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 25 2020

T:= proc(n, k) option remember;
      if k=0 then 1
    elif k=n and n<4 then 1
    elif k=n then 0
    else T(n-1, k) + T(n-2, k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Jan 25 2020

T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==n && n<4, 1, If[k==n, 0, T[n-1, k] + T[n-2, k-1] ]]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 25 2020 *)

T(n,k) = if(k==0, 1, if(k==n && n<4, 1, if(k==n, 0, T(n-1, k) + T(n-2, k-1) )));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jan 25 2020

@CachedFunction
def T(n, k):
    if (k==0): return 1
    elif (k==n and n<4): return 1
    elif (k==n): return 0
    else: return T(n-1, k) + T(n-2, k-1)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 25 2020

Typo in definition corrected by Georg Fischer, Dec 03 2021

A057703 a(n) = n(94 + 5n + 25n^2 - 5n^3 + n^4)/120.

Original entry on oeis.org

0, 1, 3, 7, 15, 31, 62, 119, 218, 381, 637, 1023, 1585, 2379, 3472, 4943, 6884, 9401, 12615, 16663, 21699, 27895, 35442, 44551, 55454, 68405, 83681, 101583, 122437, 146595, 174436, 206367, 242824, 284273, 331211, 384167, 443703, 510415, 584934, 667927

Offset: 0

Leonid Broukhis, Oct 24 2000

Previous name was: This sequence is the result of the question: If you have a tall building and 5 plates and you need to find the highest story from which a plate thrown does not break, what is the number of stories you can handle given n tries?

Number of compositions with at most five parts and sum at most n. - Beimar Naranjo, Mar 12 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372. [_Parthasarathy Nambi_, Sep 30 2009]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A004006.

Differences form A055795 + 1 = A000127.

List([0..40], n-> n*(94+5*n+25*n^2-5*n^3+n^4)/120); # G. C. Greubel, Jun 05 2019

[n*(94+5*n+25*n^2-5*n^3+n^4)/120: n in [0..40]]; // G. C. Greubel, Jun 05 2019

seq(sum(binomial(n,k),k=1..5),n=0..38); # Zerinvary Lajos, Dec 13 2007

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 3, 7, 15, 31}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

vector(40, n, n--; n*(94+5*n+25*n^2-5*n^3+n^4)/120) \\ G. C. Greubel, Jun 05 2019

[n*(94+5*n+25*n^2-5*n^3+n^4)/120 for n in (0..40)] # G. C. Greubel, Jun 05 2019

a(n) = n*(94 + 5*n + 25*n^2 - 5*n^3 + n^4)/120.
a(n) = Sum_{j=1..5} binomial(n, j). - Labos Elemer
G.f.: x*(1 - 3*x + 4*x^2 - 2*x^3 + x^4)/(1-x)^6. - Colin Barker, Apr 15 2012
E.g.f.: x*(120 + 60*x + 20*x^2 + 5*x^3 + x^4)*exp(x)/120. - G. C. Greubel, Jun 05 2019

More terms and formula from James Sellers, Oct 25 2000

Name changed by G. C. Greubel, Jun 06 2019

A227385 T(n,k)=Number of nXk 0,1 arrays indicating 2X2 subblocks of some larger (n+1)X(k+1) binary array having a sum of one, with rows and columns of the latter in lexicographically nondecreasing order.

Original entry on oeis.org

2, 3, 3, 4, 7, 4, 5, 15, 15, 5, 6, 30, 54, 30, 6, 7, 56, 185, 185, 56, 7, 8, 98, 587, 1104, 587, 98, 8, 9, 162, 1704, 6160, 6160, 1704, 162, 9, 10, 255, 4532, 31073, 61127, 31073, 4532, 255, 10, 11, 385, 11126, 141192, 550010, 550010, 141192, 11126, 385, 11, 12, 561

Offset: 1

R. H. Hardin Jul 09 2013

Table starts
..2...3.....4.......5.........6...........7...........8...........9..........10
..3...7....15......30........56..........98.........162.........255.........385
..4..15....54.....185.......587........1704........4532.......11126.......25430
..5..30...185....1104......6160.......31073......141192......581706.....2192737
..6..56...587....6160.....61127......550010.....4450124....32473856...215116595
..7..98..1704...31073....550010.....8988949...133142369..1779353333.21501389691
..8.162..4532..141192...4450124...133142369..3657501287.91016881301
..9.255.11126..581706..32473856..1779353333.91016881301
.10.385.25430.2192737.215116595.21501389691

			Some solutions for n=4 k=4
..1..0..0..0....0..0..0..1....1..0..0..0....0..0..1..0....0..0..1..0
..0..0..1..0....1..0..0..0....0..0..1..0....0..1..0..1....0..1..0..0
..0..0..1..1....0..0..0..0....0..0..1..0....1..0..1..1....1..0..1..0
..0..0..1..1....0..0..0..1....0..0..0..0....0..1..0..1....0..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..111

Column 2 is A055795(n+2)

Empirical for column k:
k=1: a(n) = n + 1
k=2: a(n) = (1/24)*n^4 + (1/12)*n^3 + (11/24)*n^2 + (17/12)*n + 1
k=3: [polynomial of degree 9] for n>3
k=4: [polynomial of degree 19] for n>7
k=5: [polynomial of degree 39] for n>22

A300401 Array T(n,k) = n(binomial(k, 2) + 1) + k(binomial(n, 2) + 1) read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 7, 8, 7, 4, 5, 11, 14, 14, 11, 5, 6, 16, 22, 24, 22, 16, 6, 7, 22, 32, 37, 37, 32, 22, 7, 8, 29, 44, 53, 56, 53, 44, 29, 8, 9, 37, 58, 72, 79, 79, 72, 58, 37, 9, 10, 46, 74, 94, 106, 110, 106, 94, 74, 46, 10, 11, 56, 92, 119

Offset: 0

Franck Maminirina Ramaharo, Mar 05 2018

Antidiagonal sums are given by 2*A055795.
Rows/columns n are binomial transform of {n, A152947(n+1), n, 0, 0, 0, ...}.
Some primes in the array are
n = 1: {2, 7, 11, 29, 37, 67, 79, 137, 191, 211, 277, 379, ...} = A055469, primes of the form k*(k + 1)/2 + 1;
n = 3: {3, 7, 37, 53, 479, 653, 1249, 1619, 2503, 3727, 4349, 5737, 7109, 8179, 9803, 11839, 12107, ...};
n = 4: {11, 37, 79, 137, 211, 821, 991, 1597, 1831, 2081, 2347, ...} = A188382, primes of the form 8*(2*k - 1)^2 + 2*(2*k - 1) + 1.

			The array T(n,k) begins
0     1    2    3    4     5     6     7     8     9    10    11  ...
1     2    4    7   11    16    22    29    37    46    56    67  ...
2     4    8   14   22    32    44    58    74    92   112   134  ...
3     7   14   24   37    53    72    94   119   147   178   212  ...
4    11   22   37   56    79   106   137   172   211   254   301  ...
5    16   32   53   79   110   146   187   233   284   340   401  ...
6    22   44   72  106   146   192   244   302   366   436   512  ...
7    29   58   94  137   187   244   308   379   457   542   634  ...
8    37   74  119  172   233   302   379   464   557   658   767  ...
9    46   92  147  211   284   366   457   557   666   784   911  ...
10   56  112  178  254   340   436   542   658   784   920  1066  ...
11   67  134  212  301   401   512   634   767   911  1066  1232  ...
12   79  158  249  352   467   594   733   884  1047  1222  1409  ...
13   92  184  289  407   538   682   839  1009  1192  1388  1597  ...
14  106  212  332  466   614   776   952  1142  1346  1564  1796  ...
15  121  242  378  529   695   876  1072  1283  1509  1750  2006  ...
16  137  274  427  596   781   982  1199  1432  1681  1946  2227  ...
17  154  308  479  667   872  1094  1333  1589  1862  2152  2459  ...
18  172  344  534  742   968  1212  1474  1754  2052  2368  2702  ...
19  191  382  592  821  1069  1336  1622  1927  2251  2594  2956  ...
20  211  422  653  904  1175  1466  1777  2108  2459  2830  3221  ...
...
The inverse binomial transforms of the columns are
0     1    2    3    4     5     6     7     8     9    10    11  ...  A001477
1     1    2    4    7    11    22    29    37    45    56    67  ...  A152947
0     1    2    3    4     5     6     7     8     9    10    11  ...  A001477
0     0    0    0    0     0     0     0     0     0     0     0  ...
0     0    0    0    0     0     0     0     0     0     0     0  ...
0     0    0    0    0     0     0     0     0     0     0     0  ...
...

Miklós Bóna, Introduction to Enumerative Combinatorics, McGraw-Hill, 2007.
L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel Publishing Company, 1974.
R. P. Stanley, Enumerative Combinatorics, second edition, Cambridge University Press, 2011.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.

Cf. A000124, A001477, A006000, A008815, A014206, A051601, A055469, A077028, A081436, A084849, A131074, A134394, A139600, A141387, A179000, A188377, A188382, A273465.

T := (n, k) -> n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1);
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

T[n_, k_] := n (Binomial[k, 2] + 1) + k (Binomial[n, 2] + 1);
Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2018 *)

T(n, k) := n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1)$
for n:0 thru 20 do
  print(makelist(T(n, k), k, 0, 20));

T(n, k) = n*(binomial(k,2) + 1) + k*(binomial(n,2) + 1);
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 12 2018

T(n,k) = T(k,n) = n*A152947(k+1) + k*A152947(n+1).
T(n,0) = A001477(n).
T(n,1) = A000124(n).
T(n,2) = A014206(n).
T(n,3) = A273465(3*n+2).
T(n,4) = A084849(n+1).
T(n,n) = A179000(n-1,n), n >= 1.
T(2*n,2*n) = 8*A081436(n-1), n >= 1.
T(2*n+1,2*n+1) = 2*A006000(2*n+1).
T(n,n+1) = A188377(n+3).
T(n,n+2) = A188377(n+2), n >= 1.
Sum_{k=0..n} T(k,n-k) = 2*(binomial(n, 4) + binomial(n, 2)).
G.f.: -((2*x*y - y - x)*(2*x*y - y - x + 1))/(((x - 1)*(y - 1))^3).
E.g.f.: (1/2)*(x + y)*(x*y + 2)*exp(x + y).

cp's OEIS Frontend

A057145 Square array of polygonal numbers T(n,k) = ((n-2)*k^2 - (n-4)*k)/2, n >= 2, k >= 1, read by antidiagonals upwards.

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A004006 a(n) = C(n,1) + C(n,2) + C(n,3), or n*(n^2 + 5)/6.

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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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A000127 Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.

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A051601 Rows of triangle formed using Pascal's rule except we begin and end the n-th row with n.

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A057145 Square array of polygonal numbers T(n,k) = ((n-2)k^2 - (n-4)k)/2, n >= 2, k >= 1, read by antidiagonals upwards.

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

A057703 a(n) = n(94 + 5n + 25n^2 - 5n^3 + n^4)/120.

A300401 Array T(n,k) = n(binomial(k, 2) + 1) + k(binomial(n, 2) + 1) read by antidiagonals.