A011379 -id:A011379 - cp's OEIS frontend

A303942 Number of partitions of n into at most 1 copy of 1^2, 2 copies of 2^2, 3 copies of 3^2, ... .

1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 0, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 3, 2, 1, 2, 2, 1, 2, 3, 4, 3, 3, 2, 2, 2, 2, 4, 4, 4, 3, 4, 3, 2, 3, 5, 7, 5, 5, 5, 6, 4, 3, 6, 8, 8, 5, 6, 6, 6, 6, 7, 9, 9, 10, 8, 8, 7, 8, 10, 11, 12, 10, 11, 10, 10, 9, 12, 15, 14, 14

Offset: 0

Seiichi Manyama, May 03 2018

			   n |              | a(n)
-----+--------------+------
   1 | 1            |  1
   4 | 4            |  1
   5 | 4+1          |  1
   8 | 4+4          |  1
   9 | 9, 4+4+1     |  2
  10 | 9+1          |  1
  13 | 9+4          |  1
  14 | 9+4+1        |  1
  16 | 16           |  1
  17 | 16+1, 9+4+4  |  2
  18 | 9+9, 9+4+4+1 |  2

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A011379, A033461, A052335, A303944, A303947.

G.f.: Product_{k>=1} (1-x^(k^2*(k+1)))/(1-x^(k^2)).

A094930 Triangle T(n,m) read by rows, defined by squaring a matrix with row entries 2+3*(m-1).

Original entry on oeis.org

4, 14, 25, 30, 65, 64, 52, 120, 152, 121, 80, 190, 264, 275, 196, 114, 275, 400, 462, 434, 289, 154, 375, 560, 682, 714, 629, 400, 200, 490, 744, 935, 1036, 1020, 860, 529, 252, 620, 952, 1221, 1400, 1462, 1380, 1127, 676, 310, 765, 1184, 1540, 1806, 1955, 1960

Offset: 1

Gary W. Adamson, Jun 17 2004

Matrix square of the matrix B(n,m) = 2+3*(m-1), B containing the first terms of A016789

in its row n, n>0, 1<=m<=n.

			The matrix B starts as
  2 ;
  2,5 ;
  2,5,8 ;
  2,5,8,11 ;
  2,5,8,11,14 ;
and interpreting this as a lower triangular matrix, its square T = B^2 starts
  4;
  14,25;
  30,65,64;
  52,120,152,121;

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Cf. A024212, A096037, A011379, A002411, A096038, A095794.

A094930 := proc(n,m) (3*m-1)*(3*m+3*n-2)*(n+1-m)/2 ; end: seq(seq(A094930(n,m),m=1..n),n=1..20) ; # R. J. Mathar, Oct 09 2009

T(n,m) = sum_{k=m..n} B(n,k)*B(k,m) = (3*m-1)*(3*m+3*n-2)*(n+1-m)/2.
Row sums: sum_{m=1..n} T(n,m) = A024212(n).
G.f. as triangle: x*y*(4+2*x+13*x*y-16*x^2*y+x^2*y^2-4*x^3*y^2)/((1-x)*(1-x*y))^3. - Robert Israel, May 06 2019

Edited and extended by R. J. Mathar, Oct 09 2009

A096037 Triangle T(n,m) = (3n+3m-2)*(n+1-m)/2 read by rows.

Original entry on oeis.org

2, 7, 5, 15, 13, 8, 26, 24, 19, 11, 40, 38, 33, 25, 14, 57, 55, 50, 42, 31, 17, 77, 75, 70, 62, 51, 37, 20, 100, 98, 93, 85, 74, 60, 43, 23, 126, 124, 119, 111, 100, 86, 69, 49, 26, 155, 153, 148, 140, 129, 115, 98, 78, 55, 29, 187, 185, 180, 172, 161, 147, 130, 110, 87, 61, 32

Offset: 1

Gary W. Adamson, Jun 17 2004

			The triangle starts in row n=1 as
2;
7,5;
15,13,8;
26,24,19,11;

Cf. A094930, A024212, A011379, A002411, A096038, A095794.

def A096037(n,m):
    return (3*n+3*m-2)*(n+1-m)//2
print( [A096037(n,m) for n in range(20) for m in range(1,n+1)] )
# R. J. Mathar, Oct 11 2009

T(n,m) = (3*n+3*m-2)*(n+1-m)/2 .
T(n,m) = A094930(n,m)/(3*m-1).
T(n,1) = A005449(n).
T(n,n) = A016768(n-1).
Row sums: sum_{m=1..n} T(n,m) = n^2*(n+1) = A011379(n).

Edited and extended, A-numbers corrected by R. J. Mathar, Oct 11 2009

A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 12, 1, 4, 12, 36, 108, 1, 5, 20, 80, 320, 1280, 1, 6, 30, 150, 750, 3750, 18750, 1, 7, 42, 252, 1512, 9072, 54432, 326592, 1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890

Offset: 0

Thomas Wieder, Mar 20 2009

Consider the k-fold Cartesian products CP(n,k) of the vector A(n) = [1, 2, 3, ..., n].
An element of CP(n,k) is a n-tuple T_t of the form T_t = [i_1, i_2, i_3, ..., i_k] with t=1, .., n^k.
We count members T of CP(n,k) which satisfy some condition delta(T_t), so delta(.) is an indicator function which attains values of 1 or 0 depending on whether T_t is to be counted or not; the summation sum_{CP(n,k)} delta(T_t) over all elements T_t of CP produces the count.
For the triangle here we have delta(T_t) = 0 if for any two i_j, i_(j+1) in T_t one has i_j = i_(j+1): T(n,k) = Sum_{CP(n,k)} delta(T_t) = Sum_{CP(n,k)} delta(i_j = i_(j+1)).
The test on i_j > i_(j+1) generates A158498. One gets the Pascal triangle A007318 if the indicator function tests whether for any two i_j, i_(j+1) in T_t one has i_j >= i_(j+1).
Use of other indicator functions can also calculate the Bell numbers A000110, A000045 or A000108.

			Array, A(n, k) = n*(n-1)^(k-1) for n > 1, A(n, k) = 1 otherwise, begins as:
  1,  1,   1,    1,     1,      1,       1,        1,        1, ... A000012;
  1,  1,   1,    1,     1,      1,       1,        1,        1, ... A000012;
  1,  2,   2,    2,     2,      2,       2,        2,        2, ... A040000;
  1,  3,   6,   12,    24,     48,      96,      192,      384, ... A003945;
  1,  4,  12,   36,   108,    324,     972,     2916,     8748, ... A003946;
  1,  5,  20,   80,   320,   1280,    5120,    20480,    81920, ... A003947;
  1,  6,  30,  150,   750,   3750,   18750,    93750,   468750, ... A003948;
  1,  7,  42,  252,  1512,   9072,   54432,   326592,  1959552, ... A003949;
  1,  8,  56,  392,  2744,  19208,  134456,   941192,  6588344, ... A003950;
  1,  9,  72,  576,  4608,  36864,  294912,  2359296, 18874368, ... A003951;
  1, 10,  90,  810,  7290,  65610,  590490,  5314410, 47829690, ... A003952;
  1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, ............. A003953;
  1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, ............. A003954;
  1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, ............. A170732;
  ... ;
The triangle begins as:
  1
  1, 1;
  1, 2,  2;
  1, 3,  6,  12;
  1, 4, 12,  36,  108;
  1, 5, 20,  80,  320,  1280;
  1, 6, 30, 150,  750,  3750,  18750;
  1, 7, 42, 252, 1512,  9072,  54432, 326592;
  1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344;
  ...;
T(3,3) = 12 counts the triples (1,2,1), (1,2,3), (1,3,1), (1,3,2), (2,1,2), (2,1,3), (2,3,1), (2,3,2), (3,1,2), (3,1,3), (3,2,1), (3,2,3) out of a total of 3^3 = 27 triples in the CP(3,3).

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

Array rows n: A170733 (n=14), ..., A170769 (n=50).
Columns k: A000012(n) (k=0), A000027(n) (k=1), A002378(n-1) (k=2), A011379(n-1) (k=3), A179824(n) (k=4), A101362(n-1) (k=5), 2*A168351(n-1) (k=6), 2*A168526(n-1) (k=7), 2*A168635(n-1) (k=8), 2*A168675(n-1) (k=9), 2*A170783(n-1) (k=10), 2*A170793(n-1) (k=11).
Diagonals k: A055897 (k=n), A055541 (k=n-1), A373395 (k=n-2), A379612 (k=n-3).
Sums: (-1)^n*A065440(n) (signed row).
Cf. A000045, A000108, A000110, A007318, A079901, A158498.

A158497:= func< n,k | k le 1 select n^k else n*(n-1)^(k-1) >;
[A158497(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2025

A158497[n_, k_]:= If[n<2 || k==0, 1, n*(n-1)^(k-1)];
Table[A158497[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2025 *)

def A158497(n,k): return n^k if k<2 else n*(n-1)^(k-1)
print(flatten([[A158497(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 18 2025

T(n, k) = (n-1)^(k-1) + (n-1)^k = n*A079901(n-1,k-1), k > 0.

Sum_{k=0..n} T(n,k) = (n*(n-1)^n - 2)/(n-2), n > 2.

Edited by R. J. Mathar, Mar 31 2009

More terms added by G. C. Greubel, Mar 18 2025

A362007 Fourth Lie-Betti number of a path graph on n vertices.

Original entry on oeis.org

0, 0, 3, 16, 48, 107, 203, 347, 551, 828, 1192, 1658, 2242, 2961, 3833, 4877, 6113, 7562, 9246, 11188, 13412, 15943, 18807, 22031, 25643, 29672, 34148, 39102, 44566, 50573, 57157, 64353, 72197, 80726, 89978, 99992, 110808

Offset: 1

Samuel J. Bevins, Apr 05 2023

Sequence T(n,4) in A360571.

Marco Aldi and Samuel Bevins, L_oo-algebras and hypergraphs, arXiv:2212.13608 [math.CO], 2022. See page 9.
Meera Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1.
Eric Weisstein's World of Mathematics, Path Graph.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A360571 (path graph triangle), A088459 (second Lie-Betti number of path graphs), A361230.

Cf. A011379, A006002, A001105, A056106, A000027, A000332, A000217.

def A362007(n):
    values = [0,0,3]
    for i in range(4, n+1):
        result = (i**4 + 18*i**3 - 97*i**2 + 174*i - 168)/24
        values.append(int(result))
    return values

a(1) = a(2) = 0, a(3) = 3, a(n) = (n^4 + 18*n^3 - 97*n^2 + 174*n - 168)/24 for n >= 4.
a(n) = A011379(n-3) + A006002(n-4) + A001105(n-3) + A056106(n-2) + A000027(n-3) + A000332(n-3) + A000217(n-5) + A000027(n-4) for n >= 5.
From Stefano Spezia, Mar 02 2025: (Start)
G.f.: x^2*(3 + x - 2*x^2 - 3*x^3 + 3*x^4 - x^5)/(1 - x)^5.
E.g.f.: (12*(6 + 4*x + x^2) - exp(x)*(72 - 24*x - 36*x^2 - 28*x^3 - x^4))/24. (End)

a(34) and Python program corrected by Robert C. Lyons, Apr 17 2023

A368116 A(m, n) = lcm_{p in Partitions(n)} (Product_{r in p}(r + m)). Array read by ascending antidiagonals, for m, n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 12, 6, 1, 4, 36, 24, 12, 1, 5, 80, 540, 720, 60, 1, 6, 150, 960, 6480, 1440, 360, 1, 7, 252, 5250, 134400, 136080, 60480, 2520, 1, 8, 392, 1512, 315000, 537600, 8164800, 120960, 5040, 1, 9, 576, 24696, 63504, 1575000, 32256000, 24494400, 3628800, 15120

Offset: 0

Peter Luschny, Dec 12 2023

We say q is a 'm-shifted partition of n' if there is a partition p of n, p = (t1, t2, ..., tk) and q = (t1 + m, t2 + m, ..., tk + m), where m is a nonnegative integer. q is a partition of n + k*m.
Let P(n) denote the partitions of n and P_{m}(n) the m-shifted partitions of n. The product of a partition is the product of its parts, Prod(p) = p1*p2*...*pk if p = (p1, p2, ..., pk). Using this terminology, the definition can be written as A(m, n) = lcm_{p in P_{m}(n)} Prod(p).
With m = 0 the cumulative radical A048803 is computed, and with m = 1 the Hirzebruch numbers A091137.

			Array A(m, n) begins:
  [0] 1, 1,   2,     6,       12,        60,           360, ...  A048803
  [1] 1, 2,  12,    24,      720,      1440,         60480, ...  A091137
  [2] 1, 3,  36,   540,     6480,    136080,       8164800, ...  A368048
  [3] 1, 4,  80,   960,   134400,    537600,      32256000, ...
  [4] 1, 5, 150,  5250,   315000,   1575000,     330750000, ...
  [5] 1, 6, 252,  1512,    63504,   1905120,     880165440, ...
  [6] 1, 7, 392, 24696,  6914880, 532445760,  268352663040, ...
  [7] 1, 8, 576, 23040, 18247680, 145981440,  683193139200, ...
  [8] 1, 9, 810, 80190,  7217100, 844400700, 5851696851000, ...
.
Let m = 2 and n = 4. The partitions of 4 are [(4), (3,1), (2,2), (2,1,1), (1, 1, 1, 1)]. Thus A(2, 4) = lcm([6, 5*3, 4*4, 4*3*3, 3*3*3*3]) = 6480.

Cf. A048803 (m=0), A091137 (m=1), A368048 (m=2).

Columns include: A000027, A011379.

def A(m, n): return lcm(product(r + m for r in p) for p in Partitions(n))
for m in range(9): print([A(m, n) for n in range(7)])

A369415 Number A(n,k) of n X n Fishburn matrices with entries in the set {0,1,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 12, 10, 0, 1, 4, 36, 264, 122, 0, 1, 5, 80, 2052, 19632, 3346, 0, 1, 6, 150, 9280, 505764, 4606752, 196082, 0, 1, 7, 252, 30750, 5684480, 511718148, 3311447232, 23869210, 0, 1, 8, 392, 83160, 39378750, 17672135680, 2088275673636, 7202118117504, 5939193962, 0

Offset: 0

Alois P. Heinz, Jan 22 2024

Number of upper triangular n X n {0,1,...,k}-matrices with no zero rows or columns.

			A(2,3) = 3*3*4 = 36:
  [10] [10] [10]  [20] [20] [20]  [30] [30] [30]
  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
.
  [11] [11] [11]  [21] [21] [21]  [31] [31] [31]
  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
.
  [12] [12] [12]  [22] [22] [22]  [32] [32] [32]
  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
.
  [13] [13] [13]  [23] [23] [23]  [33] [33] [33]
  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
.
Square array A(n,k) begins:
  1,    1,       1,         1,           1,            1, ...
  0,    1,       2,         3,           4,            5, ...
  0,    2,      12,        36,          80,          150, ...
  0,   10,     264,      2052,        9280,        30750, ...
  0,  122,   19632,    505764,     5684480,     39378750, ...
  0, 3346, 4606752, 511718148, 17672135680, 305416893750, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..53, flattened
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
Vít Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614; arXiv preprint, arXiv:1106.2261 [math.CO], 2011.
Wikipedia, Peter C. Fishburn

Columns k=0-3 give: A000007, A005321, A289314, A289315.
Rows n=0-3 give: A000012, A001477, A011379, A369423.
Main diagonal gives A369336.

A:= (n, k)-> coeff(series(add(x^j*mul(((k+1)^i-1)/(1+x*
           ((k+1)^i-1)), i=1..j), j=0..n), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

A(n,k) = [x^n] Sum_{j=0..n} x^j * Product_{i=1..j} ((k+1)^i-1)/(1+x*((k+1)^i-1)).

A132998 a(n) = n^4 - n^3 - n^2.

Original entry on oeis.org

0, -1, 4, 45, 176, 475, 1044, 2009, 3520, 5751, 8900, 13189, 18864, 26195, 35476, 47025, 61184, 78319, 98820, 123101, 151600, 184779, 223124, 267145, 317376, 374375, 438724, 511029, 591920, 682051, 782100, 892769

Offset: 0

Omar E. Pol, Nov 01 2007

			a(7)=2009 because 7^4=2401, 7^3=343, 7^2=49 and we can write 2401-343-49=2009.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A000578, A000583, A011379, A045991.

[n^4-n^3-n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

A132998:=n->n^4-n^3-n^2; seq(A132998(n), n=0..50); # Wesley Ivan Hurt, May 21 2014

f[n_]:=n^4-n^3-n^2; Table[f[n],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Dec 04 2009 *)
CoefficientList[Series[- x (-1 + 9 x + 15 x^2 + x^3)/(-1 + x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 21 2014 *)
LinearRecurrence[{5,-10,10,-5,1}, {0, -1, 4, 45, 176}, 50] (* G. C. Greubel, Sep 28 2017 *)

x='x+O('x^50); Vec(x*(-1+9*x+15*x^2+x^3)/(1-x)^5) \\ G. C. Greubel, Sep 28 2017

a(n) = n^4 - n^3 - n^2.

G.f.: x*(-1+9*x+15*x^2+x^3)/(1-x)^5. - R. J. Mathar, Nov 14 2007

A202195 Number of (n+2) X 3 binary arrays avoiding patterns 001 and 101 in rows and columns.

Original entry on oeis.org

108, 240, 450, 756, 1176, 1728, 2430, 3300, 4356, 5616, 7098, 8820, 10800, 13056, 15606, 18468, 21660, 25200, 29106, 33396, 38088, 43200, 48750, 54756, 61236, 68208, 75690, 83700, 92256, 101376, 111078, 121380, 132300, 143856, 156066, 168948

Offset: 1

R. H. Hardin, Dec 14 2011

nonn

Column 1 of A202202.

			Some solutions for n=10:
  0 0 0    0 0 0    1 0 0    1 0 0    1 0 0    0 1 1    0 1 1
  1 1 1    0 1 1    0 1 1    1 1 0    1 1 1    1 1 1    1 1 1
  1 1 0    0 1 1    0 1 1    0 1 0    1 1 1    1 1 1    1 1 1
  1 1 0    0 1 1    0 1 1    0 1 0    1 1 0    1 1 1    1 1 1
  1 1 0    0 1 1    0 1 0    0 1 0    1 0 0    1 1 1    1 1 1
  0 1 0    0 1 1    0 0 0    0 1 0    0 0 0    1 1 1    1 1 0
  0 1 0    0 1 1    0 0 0    0 1 0    0 0 0    1 1 0    1 0 0
  0 1 0    0 1 1    0 0 0    0 0 0    0 0 0    1 1 0    0 0 0
  0 1 0    0 1 0    0 0 0    0 0 0    0 0 0    0 1 0    0 0 0
  0 1 0    0 1 0    0 0 0    0 0 0    0 0 0    0 1 0    0 0 0
  0 1 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 0    0 0 0

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A202202.

Empirical: a(n) = 3*(n+3)*(n+2)^2 = 3*A011379(n+2).
Conjectures from Colin Barker, Mar 03 2018: (Start)
G.f.: 6*x*(18 - 32*x + 23*x^2 - 6*x^3) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
(End)

A286933 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - kx/(1 - kx^2/(1 - kx^3/(1 - kx^4/(1 - k*x^5/(1 - ...)))))).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 2, 0, 1, 4, 9, 12, 3, 0, 1, 5, 16, 36, 32, 5, 0, 1, 6, 25, 80, 135, 88, 9, 0, 1, 7, 36, 150, 384, 513, 248, 15, 0, 1, 8, 49, 252, 875, 1856, 1971, 688, 26, 0, 1, 9, 64, 392, 1728, 5125, 9024, 7533, 1920, 45, 0, 1, 10, 81, 576, 3087, 11880, 30125, 43776, 28836, 5360, 78, 0

Offset: 0

Ilya Gutkovskiy, May 16 2017

			G.f. of column k: A(x) = 1 + k*x + k^2*x^2 + k^2*(k + 1)*x^3 + k^3*(k + 2)*x^4 + k^3*(k^2 + 3*k + 1)*x^5 + ...
Square array begins:
  1,  1,   1,    1,     1,     1,  ...
  0,  1,   2,    3,     4,     5,  ...
  0,  1,   4,    9,    16,    25,  ...
  0,  2,  12,   36,    80,   150,  ...
  0,  3,  32,  135,   384,   875,  ...
  0,  5,  88,  513,  1856,  5125,  ...

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Columns k=0..1 give: A000007, A005169.
Rows n=0..3 give: A000012, A001477, A000290, A011379.
Main diagonal gives A291274.
Cf. A286932.

Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-k x^i, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: 1/(1 - k*x/(1 - k*x^2/(1 - k*x^3/(1 - k*x^4/(1 - k*x^5/(1 - ...)))))), a continued fraction.

G.f. of column k (for k > 0): (Sum_{j>=0} (-k)^j*x^(j*(j+1))/Product_{i=1..j} (1 - x^i)) / (Sum_{j>=0} (-k)^j*x^(j^2)/Product_{i=1..j} (1 - x^i)).

cp's OEIS Frontend

A303942 Number of partitions of n into at most 1 copy of 1^2, 2 copies of 2^2, 3 copies of 3^2, ... .

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A094930 Triangle T(n,m) read by rows, defined by squaring a matrix with row entries 2+3*(m-1).

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Maple

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A096037 Triangle T(n,m) = (3*n+3*m-2)*(n+1-m)/2 read by rows.

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Python

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A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.

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Magma

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SageMath

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A362007 Fourth Lie-Betti number of a path graph on n vertices.

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A368116 A(m, n) = lcm_{p in Partitions(n)} (Product_{r in p}(r + m)). Array read by ascending antidiagonals, for m, n >= 0.

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SageMath

A369415 Number A(n,k) of n X n Fishburn matrices with entries in the set {0,1,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A132998 a(n) = n^4 - n^3 - n^2.

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A096037 Triangle T(n,m) = (3n+3m-2)*(n+1-m)/2 read by rows.

A286933 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - kx/(1 - kx^2/(1 - kx^3/(1 - kx^4/(1 - k*x^5/(1 - ...)))))).