A077028 -id:A077028 - cp's OEIS frontend

A374452 Iterated rascal triangle R3: T(n,k) = Sum_{m=0..3} binomial(n-k,m)*binomial(k,m).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 69, 56, 28, 8, 1, 1, 9, 36, 84, 121, 121, 84, 36, 9, 1, 1, 10, 45, 120, 195, 226, 195, 120, 45, 10, 1

Offset: 0

Kolosov Petro, Jul 08 2024

Triangle T(n,k) is the third triangle R3 among the rascal-family triangles; A077028 is triangle R1, A374378 is triangle R2.
Triangle T(n,k) equals Pascal's triangle A007318 through row 2i+1, i=2 (i.e., row 7).
Triangle T(n,k) equals Pascal's triangle A007318 through column i, i=2 (i.e., column 3).

			Triangle begins:
--------------------------------------------------
k=     0   1   2   3    4    5    6   7   8   9 10
--------------------------------------------------
n=0:   1
n=1:   1   1
n=2:   1   2   1
n=3:   1   3   3   1
n=4:   1   4   6   4    1
n=5:   1   5  10  10    5    1
n=6:   1   6  15  20   15    6    1
n=7:   1   7  21  35   35   21    7   1
n=8:   1   8  28  56   69   56   28   8   1
n=9:   1   9  36  84  121  121   84  36   9   1
n=10:  1  10  45 120  195  226  195  120  45  10  1

Jena Gregory, Brandt Kronholm and Jacob White, Iterated rascal triangles, Aequationes mathematicae, 2023.
Jena Gregory, Iterated rascal triangles, Theses and Dissertations. 1050., The University of Texas Rio Grande Valley, 2022.
Amelia Gibbs and Brian K. Miceli, Two Combinatorial Interpretations of Rascal Numbers, arXiv:2405.11045 [math.CO], 2024.
Philip K. Hotchkiss, Student Inquiry and the Rascal Triangle, arXiv:1907.07749 [math.HO], 2019.
Philip K. Hotchkiss, Generalized Rascal Triangles, Journal of Integer Sequences, Vol. 23, 2020.
Petro Kolosov, Identities in Iterated Rascal Triangles, 2024.

Cf. A077028, A000027, A000217, A000292, A005894, A247608, A008860, A374378, A007318, A096943, A096940

t[n_, k_] := Sum[Binomial[n - k, m]*Binomial[k, m], {m, 0, 3}]; Column[Table[t[n, k], {n, 0, 12}, {k, 0, n}], Left]

T(n,k) = 1 + k*(n-k) + 1/4*(k-1)*k*(n-k-1)*(n-k) + 1/36*(k-2)*(k-1)*k*(n-k-2)*(n-k-1)*(n-k).
Row sums give A008860(n).
Diagonal T(n+1, n) gives A000027(n).
Diagonal T(n+2, n) gives A000217(n).
Diagonal T(n+3, n) gives A000292(n).
Diagonal T(n+4, n) gives A005894(n).
Diagonal T(n+6, n) gives A247608(n).
Column k=4 difference binomial(n+8, 4) - T(n+8, 4) gives C(n+4,4)=A007318(n+4,4).
Column k=5 difference binomial(n+9, 5) - T(n+9, 5) gives sixth column of (1,5)-Pascal triangle A096943.
G.f.: (1 + 4*x^6*y^3 - 3*x*(1 + y) - 6*x^5*y^2*(1 + y) + 2*x^4*y*(2 + 7*y+ 2*y^2) + x^2*(3 + 10*y + 3*y^2) - x^3*(1 + 11*y + 11*y^2 + y^3))/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Jul 09 2024

A077029 Rectangle R(i,j) read by ascending antidiagonals: column j has j-1 zeros followed by numbers congruent to 1 mod j-1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 0, 0, 1, 5, 5, 1, 0, 0, 0, 1, 6, 7, 4, 0, 0, 0, 0, 1, 7, 9, 7, 1, 0, 0, 0, 0, 1, 8, 11, 10, 5, 0, 0, 0, 0, 0, 1, 9, 13, 13, 9, 1, 0, 0, 0, 0, 0, 1, 10, 15, 16, 13, 6, 0, 0, 0, 0, 0, 0, 1, 11, 17, 19, 17, 11, 1, 0, 0, 0, 0, 0, 0, 1, 12, 19, 22, 21, 16, 7, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling, Oct 19 2002

The number of occurrences of k is the number of divisors of k-1, for k>=2. Column 1 of the inverse of this infinite matrix begins with (1,-1,1,-1,2,-6,24,-120,6!,-7!,8!,-9!,...). The remaining columns are likewise expressible in terms of factorials.

			Northwest corner:
1 0 0 0 0 0
1 1 0 0 0 0
1 2 1 0 0 0
1 3 3 1 0 0
1 4 5 4 1 0
1 5 7 7 5 1

Cf. A077028.

R(i, j) = (i-j)*(j-1)+1 if 1<=j<=i; R(i, j)=0 if j>=i+1.

Definition amended by Georg Fischer, Oct 25 2021

A107616 Triangle read by rows, generated from arithmetic sequences.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 16, 27, 15, 1, 6, 25, 64, 77, 26, 1, 7, 36, 125, 247, 207, 42, 1, 8, 49, 216, 609, 916, 529, 64

Offset: 1

Gary W. Adamson, May 17 2005

			Row 4 of A077028 is [1, 3, 3, 1] which becomes f(x): x^3 + 3x^2 + 3x + 1. Given x = 1,2,3...; f(x) generates 8, 27, 64, 125, ... (4th offset column of A107616).
First few rows of the triangle are:
  1;
  1, 2;
  1, 3, 4;
  1, 4, 9, 8;
  1, 5, 16, 27, 15;
  1, 6, 25, 64, 77, 26;
  ...

Cf. A077028.

Row terms of A077028 are considered as coefficients to generating functions such that n-th row of A077028 becomes the generator for n-th column of A107616.

A140348 Growth function for the submonoid generated by the generators of the free nil-2 group on three generators.

Original entry on oeis.org

1, 3, 9, 27, 78, 216, 568, 1410, 3309, 7307, 15303

Offset: 0

David S. Newman and Moshe Shmuel Newman, May 29 2008

The process of expressing a word in generators as a sorted word in generators and commutators is Marshall Hall's 'collection process'.
Since this monoid 'lives in' a nilpotent group, it inherits the growth restriction of a nilpotent group. So according to a result of Bass, a(n) = O( n^8).
It seems this is the correct growth rate. This sequence may well have a rational generating function, though, according to a result of M Stoll, the growth function of a nilpotent group need not be rational, or even algebraic.
Computations on a free nilpotent group, or on submonoids, may be aided by using matricies. I. D. MacDonald describes how to do this in an American Mathematical Monthly article and he gives a recipe explicitly for the nil-2, 3 generator case.

			Suppose the generators are a,b,c and their commutators are q,r,s, so:
ba = abq, ca = acr, cb = bcs;
nil-2 means that q,r,s commute with everything.
Now there are 81 different words of length 4 on a,b,c, but there are three equations:
abba = baab ( = aabbqq)
acca = caac ( = aaccrr)
bccb = cbbc ( = bbccss)
and these are the only equations, so instead of 81 distinct words we have 78 distinct words, a(4)=78.

H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. 25 (1972).
I. D. MacDonald, Commutators and Their Products, The American Mathematical Monthly, Vol. 93, No. 6, (Jun. - Jul., 1986), pp. 440-444.
Michael Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Inventiones Mathematicae Volume 126, Number 1 / September, 1996.

Cf. sequence A000125 gives the analogous count for the 2 generator case. sequence A077028 refines A000125 by giving the number of words with k a's and (n-k)b's.

A296180 Triangle read by rows: T(n, k) = 3(n - k)k + 1, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 13, 10, 1, 1, 13, 19, 19, 13, 1, 1, 16, 25, 28, 25, 16, 1, 1, 19, 31, 37, 37, 31, 19, 1, 1, 22, 37, 46, 49, 46, 37, 22, 1, 1, 25, 43, 55, 61, 61, 55, 43, 25, 1, 1, 28, 49, 64, 73, 76, 73, 64, 49, 28, 1

Offset: 0

Wolfdieter Lang, Dec 20 2017

This is member m = 3 of the family of triangles T(m; n, k) = m*(n - k)*k + 1, for m >= 0. For m = 0: A000012(n, k) (read as a triangle); for m = 1: A077028 (rascal), for m = 2: T(2, n+1, k+1) = A130154(n, k). Motivated by A130154 to look at this family of triangles.
In general the recurrence is: T(m; n, 0) = 1 and T(m; n, n) = 1 for n >= 0; T(m; n, k) = (T(m; n-1, k-1)*T(m; n-1, k) + m)/T(m; n-2, k-1), for n >= 2, k = 1..n-1.
The general g.f. of the sequence of column k (with leading zeros) is G(m; k, x) = (x^k/(1 - x)^2)*(1 + (m*k - 1)*x), k >= 0.
The general g.f. of the triangle T(m;, n, k) is GT(m; x, t) = (1 - (1 + t)*x + (m+1)*t*x^2)/((1 - t*x)*(1 - x))^2, and G(m; k, x) = (d/dt)^k GT(m; x, t)/k!|_{t=0}.
For a simple combinatorial interpretation see the one given in A130154 by Rogério Serôdio which can be generalized to m >= 3.

			The triangle T(n, k) begins:
n\k   0  1  2  3  4  5  6  7  8  9 10 ...
0:    1
1:    1  1
2:    1  4  1
3:    1  7  7  1
4:    1 10 13 10  1
5:    1 13 19 19 13  1
6:    1 16 25 28 25 16  1
7:    1 19 31 37 37 31 19  1
8:    1 22 37 46 49 46 37 22  1
9:    1 25 43 55 61 61 55 43 25  1
10:   1 28 49 64 73 76 73 64 49 28  1
...
Recurrence: 28 = T(6, 3) = (19*19 + 3)/13 = 28.

Cf. A077028, A130154.

Columns (without leading zeros): A000012, A016777, A016921, A016921, A017173, A017533, ...

Table[3 k (n - k) + 1, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 20 2017 *)

lista(nn) = for(n=0, nn, for(k=0, n, print1(3*(n - k)*k + 1, ", "))) \\ Iain Fox, Dec 21 2017

T(n, k) = 3*(n - k)*k + 1, n >= 0, 0 <= k <= n,
Recurrence: T(n, 0) = 1 and T(n, n) = 1 for n >= 0; T(n, k) = (T(n-1, k-1)*T(n-1, k) + 3)/T(n-2, k-1), for n >= 2, k = 1..n-1.
G.f. of column k (with leading zeros): (x^k/(1 - x)^2)*(1 + (3*k-1)*x), k >= 0.
G.f. of triangle: (1 - (1 + t)*x + 4*t*x^2)/((1 - t*x)*(1 - x))^2 = 1 + (1+t)*x +(1 + 4*t + t^2)*x^2 + (1 + 7*t + 7*t^2 + t^3)*x^3 = ...

A306326 The q-analogs T(q; n,k) of the rascal-triangle, here q = 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 10, 8, 1, 1, 16, 22, 22, 16, 1, 1, 32, 46, 50, 46, 32, 1, 1, 64, 94, 106, 106, 94, 64, 1, 1, 128, 190, 218, 226, 218, 190, 128, 1, 1, 256, 382, 442, 466, 466, 442, 382, 256, 1, 1, 512, 766, 890, 946, 962, 946, 890, 766, 512, 1

Offset: 0

Werner Schulte, Feb 07 2019

The formulas are given for the general case depending on some fixed integer q. The terms are valid if q = 2. For the special case q = 1 see A077028, for q = 3 see A306344. For q < 1 the terms might be negative.

			If q = 2 the triangle T(2; n,k) starts:
n\k:  0     1     2     3     4     5     6     7     8     9
=============================================================
  0:  1
  1:  1     1
  2:  1     2     1
  3:  1     4     4     1
  4:  1     8    10     8     1
  5:  1    16    22    22    16     1
  6:  1    32    46    50    46    32     1
  7:  1    64    94   106   106    94    64     1
  8:  1   128   190   218   226   218   190   128     1
  9:  1   256   382   442   466   466   442   382   256     1
etc.

Cf. A077028, A306344.

T(q; n,k) = 1 + ((q^k-1)/(q-1))*(q^(n-k)-1)/(q-1) for 0 <= k <= n.
T(q; n,k) = T(q; n,n-k) for 0 <= k <= n.
T(q; n,0) = T(q; n,n) = 1 for n >= 0.
T(q; n,1) = 1 + (q^(n-1)-1)/(q-1) for n > 0.
T(q; i,j) = 0 if i < j or j < 0.
The T(q; n,k) satisfy several recurrence equations:
  (1) T(q; n,k) = q*T(q; n-1,k) + (q^k-1)/(q-1)-(q-1) for 0 <= k < n;
  (2) T(q; n,k) = (T(q; n-1,k)*T(q; n-1,k-1) + q^(n-2))/T(q; n-2,k-1),
  (3) T(q; n,k) = T(q; n,k-1) + T(q; n-1,k) + q^(n-k-1) - T(q; n-1,k-1),
  (4) T(q; n,k) = T(q; n,k-1) + q*T(q; n-2,k-1) - q*T(q; n-2,k-2) for 0 < k < n;
  (5) T(q; n,k) = T(q; n,k-2) + T(q; n-1,k) + (1+q)*q^(n-k-1) - T(q; n-1,k-2)
  for 1 < k < n with initial values given above.
G.f. of column k >= 0: Sum_{n>=0} T(q; n+k,k)*t^n = (1+((q^k-1)/(q-1)-q)*t) / ((1-t)*(1-q*t)). Take account of lim_{q->1} (q^k-1)/(q-1) = k.
G.f.: Sum_{n>=0, k=0..n} T(q; n,k)*x^k*t^n = (1-q*t-q*x*t+(1+q^2)*x*t^2) / ((1-t)*(1-q*t)*(1-x*t)*(1-q*x*t)).
The row polynomials p(q; n,x) = Sum_{k=0..n} T(q; n,k)*x^k satisfy the recurrence equation p(q; n,x) = q*p(q; n-1,x) + x^n + Sum_{k=0..n-1} ((q^k-1)/(q-1)-(q-1))*x^k for n > 0 with initial value p(q; 0,x) = 1.

A306344 The q-analogs T(q; n,k) of the rascal-triangle, here q = 3.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 14, 17, 14, 1, 1, 41, 53, 53, 41, 1, 1, 122, 161, 170, 161, 122, 1, 1, 365, 485, 521, 521, 485, 365, 1, 1, 1094, 1457, 1574, 1601, 1574, 1457, 1094, 1, 1, 3281, 4373, 4733, 4841, 4841, 4733, 4373, 3281, 1

Offset: 0

Werner Schulte, Feb 08 2019

The formulas are given for the general case depending on some fixed integer q. The terms are valid for q = 3. For the special case q = 1 see A077028, for q = 2 see A306326. For q < 1 the terms might be negative.

			If q = 3 the triangle T(3; n,k) starts:
n\k:  0     1     2     3     4     5     6     7     8     9
=============================================================
  0:  1
  1:  1     1
  2:  1     2     1
  3:  1     5     5     1
  4:  1    14    17    14     1
  5:  1    41    53    53    41     1
  6:  1   122   161   170   161   122     1
  7:  1   365   485   521   521   485   365     1
  8:  1  1094  1457  1574  1601  1574  1457  1094     1
  9:  1  3281  4373  4733  4841  4841  4733  4373  3281     1
etc.

Cf. A077028, A306326.

T(q; n,k) = 1 + ((q^k-1)/(q-1))*((q^(n-k)-1)/(q-1)) for 0 <= k <= n.
T(q; n,k) = T(q; n,n-k) for 0 <= k <= n.
T(q; n,0) = T(q; n,n) = 1 for n >= 0.
T(q; n,1) = 1 + (q^(n-1)-1)/(q-1) for n > 0.
T(q; i,j) = 0 if i < j or j < 0.
The T(q; n,k) satisfy several recurrence equations:
  (1) T(q; n,k) = q*T(q; n-1,k) + (q^k-1)/(q-1)-(q-1) for 0 <= k < n;
  (2) T(q; n,k) = (T(q; n-1,k)*T(q; n-1,k-1) + q^(n-2))/T(q; n-2,k-1),
  (3) T(q; n,k) = T(q; n,k-1) + T(q; n-1,k) + q^(n-k-1) - T(q; n-1,k-1),
  (4) T(q; n,k) = T(q; n,k-1) + q*T(q; n-2,k-1) - q*T(q; n-2,k-2) for 0 < k < n;
  (5) T(q; n,k) = T(q; n,k-2) + T(q; n-1,k) + (1+q)*q^(n-k-1) - T(q; n-1,k-2)
  for 1 < k < n  with initial values given above.
G.f. of column k >= 0: Sum_{n>=0} T(q; n+k,k)*t^n = (1+((q^k-1)/(q-1)-q)*t) / ((1-t)*(1-q*t)). Take account of lim_{q->1} (q^k-1)/(q-1) = k.
G.f.: Sum_{n>=0, k=0..n} T(q; n,k)*x^k*t^n = (1-q*t-q*x*t+(1+q^2)*x*t^2) / ((1-t)*(1-q*t)*(1-x*t)*(1-q*x*t)).
The row polynomials p(q; n,x) = Sum_{k=0..n} T(q; n,k)*x^k satisfy the recurrence equation p(q; n,x) = q*p(q; n-1,x) + x^n + Sum_{k=0..n-1} ((q^k-1)/(q-1)-(q-1))*x^k for n > 0 with initial value p(q; 0,x) = 1.

A332790 Triangle read by rows: T(n,k) = 1 + 2n + k + 5k(n-k) for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 3, 4, 5, 11, 7, 7, 18, 19, 10, 9, 25, 31, 27, 13, 11, 32, 43, 44, 35, 16, 13, 39, 55, 61, 57, 43, 19, 15, 46, 67, 78, 79, 70, 51, 22, 17, 53, 79, 95, 101, 97, 83, 59, 25, 19, 60, 91, 112, 123, 124, 115, 96, 67, 28, 21, 67, 103, 129, 145, 151, 147, 133, 109, 75, 31

Offset: 0

Philip K Hotchkiss, Mar 04 2020

			From _Jon E. Schoenfield_, Mar 14 2020: (Start)
.
  n\k|  0    1    2    3    4    5    6    7    8    9   10
  ---+-----------------------------------------------------
   0 |  1
   1 |  3    4
   2 |  5   11    7
   3 |  7   18   19   10
   4 |  9   25   31   27   13
   5 | 11   32   43   44   35   16
   6 | 13   39   55   61   57   43   19
   7 | 15   46   67   78   79   70   51   22
   8 | 17   53   79   95  101   97   83   59   25
   9 | 19   60   91  112  123  124  115   96   67   28
  10 | 21   67  103  129  145  151  147  133  109   75   31
  ...
(End)

Philip K. Hotchkiss, Generalized Rascal Triangles, arXiv:1907.11159 [math.HO], 2019, Figure 8 p. 3.

Cf. A077028, A309555, A309557, A309559, A332963

:=proc(n, k)
   if n<0 or k<0 or k>n then
       0;
   else
       1+2*n+k+5*k*(n-k);
   end if;

T[n_, k_]:=1+2*n+k+5*k*(n-k); Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten

T(n,k) = 1 + 2*n + k + 5*k*(n-k), n >= 0, 0 <= k <= n.

A332963 Number triangle where T(2n,0)=T(2n,2n)=1, T(2n+1,0)=T(2n+1,2n+1)=2 for all n >= 0, and the interior numbers are defined recursively by T(n,k) = (T(n-1,k-1)*T(n-1,k)+1)/T(n-2,k-1) for n > 2, 0 < k <= n.

Original entry on oeis.org

1, 2, 2, 1, 5, 1, 2, 3, 3, 2, 1, 7, 2, 7, 1, 2, 4, 5, 5, 4, 2, 1, 9, 3, 13, 3, 9, 1, 2, 5, 7, 8, 8, 7, 5, 2, 1, 11, 4, 19, 5, 19, 4, 11, 1, 2, 6, 9, 11, 12, 12, 11, 9, 6, 2, 1, 13, 5, 25, 7, 29, 7, 25, 5, 13, 1, 2, 7, 11, 14, 16, 17, 17, 16, 14, 11, 7, 2

Offset: 0

Philip K Hotchkiss, Mar 04 2020

			For row 3: a(3,0)=2, a(3,1)= 3, a(3,2)=3, a(3,3)=2.
For antidiagonal 3: T(3,0)=2, T(3,1)=7, T(3,2)=5, T(3,3)=13, ...
Triangle begins:
  1;
  2, 2;
  1, 5, 1;
  2, 3, 3, 2;
  1, 7, 2, 7, 1;
  2, 4, 5, 5, 4, 2;
  ...

Philip K. Hotchkiss, Generalized Rascal Triangles, arXiv:1907.11159 [math.HO], 2019, Figure 10 p. 4.

Cf. A077028, A309555, A309557, A332790, A309559

T(n, k) = if ((n<0) || (nMichel Marcus, Mar 16 2020

By rows: a(2n,0)=a(2n,2n)=1, a(2n+1,0)=a(2n+1,2n+1)=2 for all n >= 0, while the interior numbers are defined recursively by a(n,k) = (a(n-1,k-1)*a(n-1,k)+1)/a(n-2,k-1) for n >= 2, 0 < k <= n.

By antidiagonals: T(0,2n)=T(2n,0)=1, T(0,2n+1)=T(2n+1,0)=2 for all n >= 0, while the interior numbers are defined recursively by T(r,k) = (T(r-1,k)*(Tr,k-1)+1)/T(r-1,k-1) for r,k > 0.

A361731 Array read by descending antidiagonals. A(n, k) = hypergeom([-k, -3], [1], n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 7, 1, 1, 20, 25, 10, 1, 1, 35, 63, 46, 13, 1, 1, 56, 129, 136, 73, 16, 1, 1, 84, 231, 307, 245, 106, 19, 1, 1, 120, 377, 586, 593, 396, 145, 22, 1, 1, 165, 575, 1000, 1181, 1011, 595, 190, 25, 1, 1, 220, 833, 1576, 2073, 2076, 1585, 848, 241, 28, 1

Offset: 0

Peter Luschny, Mar 22 2023

			Array A(n, k) starts:
 [0] 1,  1,   1,   1,    1,    1,    1,     1, ...  A000012
 [1] 1,  4,  10,  20,   35,   56,   84,   120, ...  A000292
 [2] 1,  7,  25,  63,  129,  231,  377,   575, ...  A001845
 [3] 1, 10,  46, 136,  307,  586, 1000,  1576, ...  A081583
 [4] 1, 13,  73, 245,  593, 1181, 2073,  3333, ...  A081586
 [5] 1, 16, 106, 396, 1011, 2076, 3716,  6056, ...  A081588
 [6] 1, 19, 145, 595, 1585, 3331, 6049,  9955, ...  A081590
 [7] 1, 22, 190, 848, 2339, 5006, 9192, 15240, ...
.
Table T(n, k) starts:
 [0] 1;
 [1] 1,   1;
 [2] 1,   4,   1;
 [3] 1,  10,   7,    1;
 [4] 1,  20,  25,   10,    1;
 [5] 1,  35,  63,   46,   13,    1;
 [6] 1,  56, 129,  136,   73,   16,   1;
 [7] 1,  84, 231,  307,  245,  106,  19,   1;
 [8] 1, 120, 377,  586,  593,  396, 145,  22,  1;
 [9] 1, 165, 575, 1000, 1181, 1011, 595, 190, 25, 1;

Rows: A000012, A000292, A001845, A081583, A081586, A081588, A081590.
Columns: A000012, A016777, A100536.
Hypergeometric family: A000012 (m=0), A077028 (m=1), A361682 (m=2), this array (m=3).

A := (n, k) -> 1 + (((k*n - 3*n + 9)*n*k + (2*n - 9)*n + 18)*n*k)/6;
seq(print(seq(A(n, k), k = 0..7)), n = 0..7);
# Alternative:
ogf := n -> (1 + (n - 1) * x)^3 / (1 - x)^4:
ser := n -> series(ogf(n), x, 12):
row := n -> seq(coeff(ser(n), x, k), k = 0..9):
seq(print(row(n)), n = 0..9);

A(n, k) = [x^k] (1 + (n - 1) * x)^3 / (1 - x)^4.
A(n, k) = 1 + (((k*n - 3*n + 9)*n*k + (2*n - 9)*n + 18)*n*k)/6.
T(n, k) = 1 + (((k*(n - k) - 3*k + 9)*k*(n - k) + (2*k - 9)*k + 18)*k*(n - k))/6.

cp's OEIS Frontend

A374452 Iterated rascal triangle R3: T(n,k) = Sum_{m=0..3} binomial(n-k,m)*binomial(k,m).

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A077029 Rectangle R(i,j) read by ascending antidiagonals: column j has j-1 zeros followed by numbers congruent to 1 mod j-1.

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Extensions

A107616 Triangle read by rows, generated from arithmetic sequences.

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A140348 Growth function for the submonoid generated by the generators of the free nil-2 group on three generators.

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A296180 Triangle read by rows: T(n, k) = 3*(n - k)*k + 1, n >= 0, 0 <= k <= n.

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PARI

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A306326 The q-analogs T(q; n,k) of the rascal-triangle, here q = 2.

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A306344 The q-analogs T(q; n,k) of the rascal-triangle, here q = 3.

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A332790 Triangle read by rows: T(n,k) = 1 + 2*n + k + 5*k(n-k) for n >= 0, 0 <= k <= n.

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Maple

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A332963 Number triangle where T(2n,0)=T(2n,2n)=1, T(2n+1,0)=T(2n+1,2n+1)=2 for all n >= 0, and the interior numbers are defined recursively by T(n,k) = (T(n-1,k-1)*T(n-1,k)+1)/T(n-2,k-1) for n > 2, 0 < k <= n.

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A296180 Triangle read by rows: T(n, k) = 3(n - k)k + 1, n >= 0, 0 <= k <= n.

A332790 Triangle read by rows: T(n,k) = 1 + 2n + k + 5k(n-k) for n >= 0, 0 <= k <= n.