A028552 -id:A028552 - cp's OEIS frontend

A265611 a(n) = a(n-1) + floor((n-1)/2) - (-1)^n + 2 for n>=2, a(0)=1, a(1)=3.

1, 3, 4, 8, 10, 15, 18, 24, 28, 35, 40, 48, 54, 63, 70, 80, 88, 99, 108, 120, 130, 143, 154, 168, 180, 195, 208, 224, 238, 255, 270, 288, 304, 323, 340, 360, 378, 399, 418, 440, 460, 483, 504, 528, 550, 575, 598, 624, 648, 675, 700, 728, 754, 783, 810, 840

Offset: 0

Peter Luschny, Dec 17 2015

Enrique Pérez Herrero, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Peter Luschny, Looking for an interpretation, seqfan mailing list.

Cf. A002620, A005563, A028552, A052928, A132411, A198442, A217748.
Cf. A084964 and A097065, after the first 3: a(n+1) - a(n) for n>0.
Cf. A055998, after 3: a(n+1) + a(n) for n>0.
Cf. A063929: a(2*n+1) gives the second column of the triangle; for n>0, a(2*n) gives the third column.

[1] cat [(2*n*(n+6)-5*(-1)^n+5)/8: n in [1..60]]; // Bruno Berselli, Dec 18 2015

A265611 := proc(n) iquo(n+1,2); %*(%+irem(n+1,2)+2)+0^n end:
seq(A265611(n), n=0..55);

Join[{1}, Table[(2 n (n + 6) - 5 (-1)^n + 5)/8, {n, 1, 60}]] (* Bruno Berselli, Dec 18 2015 *)

Vec((x^4-2*x^3+2*x^2-x-1)/(x^4-2*x^3+2*x-1) + O(x^1000)) \\ Altug Alkan, Dec 18 2015

# The initial values x, y = 0, 1 give the quarter-squares A002620.
def A265611():
    x, y = 1, 2
    while True:
       yield x
       x, y = x + y, x//y + 1
a = A265611(); print([next(a) for i in range(60)])

O.g.f.: (x^4-2*x^3+2*x^2-x-1)/(x^4-2*x^3+2*x-1).
E.g.f.: 1-(5/8)*exp(-x)+(1/8)*(5+14*x+2*x^2)*exp(x).
a(2*n) = n*(n+3) + 0^n = A028552(n) + 0^n. [Here 0^0 = 1, otherwise 0^s = 0. - N. J. A. Sloane, Aug 26 2022]
a(2*n+1) = (n+1)*(n+3) = A005563(n+1).
a(n+1) - a(n) = floor(n/2) + 2 + (-1)^n - 0^n.
a(n) = a(-n-6) = (2*n*(n+6) - 5*(-1)^n + 5)/8 for n>0, a(0)=1. [Bruno Berselli, Dec 18 2015]
For n>0, a(n) = n + 1 + Sum_{i=1..n+1} floor(i/2) + (-1)^i =  n + floor((n+1)^2/4) + (1 - (-1)^n)/2. - Enrique Pérez Herrero, Dec 18 2015
Sum_{n>=0} 1/a(n) = 85/36. - Enrique Pérez Herrero, Dec 18 2015
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>5. - R. H. Hardin, Dec 21 2015, proved by Susanne Wienand for the algorithm sent to the seqfan mailing list and used in the Sage script below.
a(n) = A002620(n+1) + A052928(n+1) for n>=1. (Note A198442(n) = A002620(n+2) - A052928(n+2) for n>=1.) - Peter Luschny, Dec 22 2015
a(n) = (floor((n+3)/2)-1)*(ceiling((n+3)/2)+1) for n>0. - Wesley Ivan Hurt, Mar 30 2017

A277978 a(n) = 3n(n+3).

Original entry on oeis.org

0, 12, 30, 54, 84, 120, 162, 210, 264, 324, 390, 462, 540, 624, 714, 810, 912, 1020, 1134, 1254, 1380, 1512, 1650, 1794, 1944, 2100, 2262, 2430, 2604, 2784, 2970, 3162, 3360, 3564, 3774, 3990, 4212, 4440, 4674, 4914, 5160, 5412, 5670, 5934, 6204, 6480

Offset: 0

Emeric Deutsch, Nov 08 2016

For n>= 3, a(n) is the second Zagreb index of the wheel graph with n+1 vertices. The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of g.

			a(3) = 54. Indeed, the wheel graph with 4 vertices consists of 6 edges, each connecting two vertices of degree 3. Then, the second Zagreb index is 6*3*3 = 54.

Leo Tavares, Illustration: Hexagonal Stars
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A028569, A140091.

Maple
```
seq(3*n*(n+3), n = 0 .. 45);
```

A277978[n_] := 3 n (n + 3); Array[A277978, 45] (* JungHwan Min, Nov 08 2016 *)

a(n)=3*n*(n+3) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2 * A140091(n) = 3 * A028552(n) = 6 * A000096(n).
G.f.: 6*x*(2-x)/(1-x)^3
a(n) = A003154(n+1) - A003215(n-1). See Hexagonal Stars illustration. - Leo Tavares, Aug 20 2021

A285089 Rectangular array by antidiagonals: row n is the ordered sequence of numbers k that minimize |d(n+1-k) - d(k)|, where d(i) are the divisors of n.

Original entry on oeis.org

1, 4, 2, 9, 6, 3, 16, 12, 8, 10, 25, 20, 15, 18, 5, 36, 30, 24, 28, 21, 14, 49, 42, 35, 40, 32, 50, 7, 64, 56, 48, 54, 45, 66, 27, 44, 81, 72, 63, 70, 60, 84, 55, 78, 33, 100, 90, 80, 88, 77, 104, 91, 98, 65, 22, 121, 110, 99, 108, 96, 126, 112, 170, 105, 52

Offset: 1

Clark Kimberling, Apr 13 2017

Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the natural numbers, A000027.
Every prime (A000040) occurs in column 1.
Row 1: A000290 (squares)
Row 2: A002378 (oblong numbers)
Row 3: A005563
Row 4: A028552 (for n>=2)

			Taking n = 12, the divisors are 1,2,3,4,6,12, so that for k=1..6, the numbers d(n+1-k) - d(k) are 12-1, 6-2, 4-3, 3-4, 2-6, 1-12.  Thus, the number k that minimizes |d(n+1-k) - d(k)| is 1, so that 12 appears in row 1 (with the top row as row 0), consisting of numbers for which the minimal value is 1.
Northwest corner:
  1   4   9   16   25   36   49   64   81   10
  2   6   12  20   30   42   56   72   90   110
  3   8   15  24   35   48   63   80   99   120
  10  18  28  40   54   70   88   108  130  154
  5   21  32  45   60   77   96   117  140  165
  14  50  66  84   104  126  160  176  204  234
  7   27  55  91   112  135  160  187  216  247
  44  78  98  170  198  228  260  294  330  368

Clark Kimberling, Antidiagonals n = 1..60, flattened
Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A000040, A000290, A095833, A163280, A285090.

d[n_] := Divisors[n]; k[n_] := Length[d[n]]; x[n_, i_] := d[n][[i]];
a[n_] := If[OddQ[k[n]], 0, x[n, k[n]/2 + 1] - x[n, k[n]/2]]
t = Table[a[j], {j, 1, 30000}];
r[n_] := Flatten[Position[t, n]]; v[n_, k_] := r[n][[k]];
w = Table[v[n, k], {n, 0, 10}, {k, 1, 10}];
TableForm[w] (* A285089, array *)
Table[v[n - k, k], {n, 0, 60}, {k, n, 1, -1}] // Flatten (* A285089, sequence *)

row 1: k^2 for k>=1
row 2: k*(k+1) for k>=1
row 3: k*(k+2) for k>=3
row 4: k*(k+3) for k>=2
row 5: k*(k+4) for k>=3
row 6: k*(k+5) for k>=5
row 7: k*(k+6) for k>=7

A367076 Irregular triangle read by rows: T(n,k) (0 <= n, 0 <= k < 2^n). T(n,k) = -Sum_{i=0..k} A365968(n,i).

Original entry on oeis.org

0, 1, 0, 3, 4, 3, 0, 6, 10, 12, 12, 12, 10, 6, 0, 10, 18, 24, 28, 32, 34, 34, 32, 34, 34, 32, 28, 24, 18, 10, 0, 15, 28, 39, 48, 57, 64, 69, 72, 79, 84, 87, 88, 89, 88, 85, 80, 85, 88, 89, 88, 87, 84, 79, 72, 69, 64, 57, 48, 39, 28, 15, 0, 21, 40, 57, 72, 87

Offset: 0

John Tyler Rascoe, Nov 05 2023

			Triangle begins:
    k=0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
n=0:  0;
n=1:  1,  0;
n=2:  3,  4,  3,  0;
n=3:  6, 10, 12, 12, 12, 10,  6,  0;
n=4; 10, 18, 24, 28, 32, 34, 34, 32, 34, 34, 32, 28, 24, 18, 10,  0;

John Tyler Rascoe, Rows n = 0..12, flattened
Wikipedia, Blancmange curve.
Index entries for sequences related to binary expansion of n

Cf. A000217 (column k=0), A028552 (column k=1), A192021 (row sums).

Cf. A004074, A249071, A274575, A277914, A296062, A365968.

nmax=10; row[n_]:=Join[CoefficientList[Series[1/(1-x)*Sum[ i/(1+x^2^(i-1))*Product[1+x^2^j,{j,0,i-2}],{i,n}],{x,0,2^n-1}],x],{0}]; Array[row,6,0] (* Stefano Spezia, Dec 23 2023 *)

def row_gen(n):
    x = 0
    for k in range(2**n):
        b = bin(k)[2:].zfill(n)
        x += sum((-1)**(int(b[n-i])+1)*i for i in range(1,n+1))
        yield(-x)
def A367076_row_n(n): return(list(row_gen(n)))

T(n,k) = Sum_{i=0..n} abs(k + 1 - (2^i) * round((k+1)/2^i)) * i.

G.f. for n-th row: 1/(1-x) * Sum_{i=1..n} (i/(1+x^2^(i-1)) * Product_{j=0..i-2} 1 + x^2^j).

A380841 Array read by ascending antidiagonals: A(n,k) = n! * [x^n] 1/(1 - x*exp(x))^k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 21, 10, 3, 1, 0, 148, 66, 18, 4, 1, 0, 1305, 560, 141, 28, 5, 1, 0, 13806, 5770, 1380, 252, 40, 6, 1, 0, 170401, 69852, 16095, 2776, 405, 54, 7, 1, 0, 2403640, 970886, 217458, 35940, 4940, 606, 70, 8, 1, 0, 38143377, 15228880, 3335745, 533304, 70045, 8088, 861, 88, 9, 1

Offset: 0

Stefano Spezia, Feb 05 2025

			Array begins as:
  1,    1,    1,     1,     1,     1,      1, ...
  0,    1,    2,     3,     4,     5,      6, ...
  0,    4,   10,    18,    28,    40,     54, ...
  0,   21,   66,   141,   252,   405,    606, ...
  0,  148,  560,  1380,  2776,  4940,   8088, ...
  0, 1305, 5770, 16095, 35940, 70045, 124350, ...
  ...

Cf. A380843 (antidiagonal sums).
Columns k=0..4 give A000007, A006153, A377529, A377530, A379993.
Rows n=0..2 give A000012, A001477, A028552.
Main diagonal gives A380842.
A(n,n+1) gives A213643(n+1).

A[n_,k_]:=n!SeriesCoefficient[1/(1-x*Exp[x])^k,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

A(n,k) = n! * Sum_{j=0..n} j^(n-j) * binomial(j+k-1,j)/(n-j)!. - Seiichi Manyama, Feb 06 2025

A070895 Triangle read by rows where T(n+1,k)=T(n,k)+n*T(n-1,k) starting with T(n,n)=1 and T(n,k)=0 if n

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 6, 4, 1, 1, 26, 18, 8, 5, 1, 1, 76, 48, 28, 10, 6, 1, 1, 232, 156, 76, 40, 12, 7, 1, 1, 764, 492, 272, 110, 54, 14, 8, 1, 1, 2620, 1740, 880, 430, 150, 70, 16, 9, 1, 1, 9496, 6168, 3328, 1420, 636, 196, 88, 18, 10, 1, 1, 35696, 23568

Offset: 0

Henry Bottomley, May 23 2002

For n>k+1, T(n,k) is a multiple of k+2.

Eigentriangle of inverse of (-1)^(n-k)*A094587. Row sums are A187044. - Paul Barry, Mar 02 2011

			Rows start: 1; 1,1; 2,1,1; 4,3,1,1; 10,6,4,1,1; etc.
Triangle begins
1,
1, 1,
2, 1, 1,
4, 3, 1, 1,
10, 6, 4, 1, 1,
26, 18, 8, 5, 1, 1,
76, 48, 28, 10, 6, 1, 1,
232, 156, 76, 40, 12, 7, 1, 1
Production matrix begins
1, 1,
1, 0, 1,
1, 1, 0, 1,
2, 1, 1, 0, 1,
4, 3, 1, 1, 0, 1,
10, 6, 4, 1, 1, 0, 1,
26, 18, 8, 5, 1, 1, 0, 1,
76, 48, 28, 10, 6, 1, 1, 0, 1,
232, 156, 76, 40, 12, 7, 1, 1, 0, 1
Inverse begins
1,
-1, 1,
-1, -1, 1,
0, -2, -1, 1,
0, 0, -3, -1, 1,
0, 0, 0, -4, -1, 1,
0, 0, 0, 0, -5, -1, 1,
0, 0, 0, 0, 0, -6, -1, 1
- _Paul Barry_, Mar 02 2011

Columns include A000085, A000932, A059480. Right hand columns effectively include A000012 (twice), A000027, A005843, A028552. Cf. A062323 for a triangle with similar formulas.

T(n, k+1)=(T(n, k-1)-T(n-1, k))/k for 0

A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 4, 2, 0, 9, 6, 3, 0, 16, 12, 8, 4, 0, 25, 20, 15, 10, 5, 0, 36, 30, 24, 18, 12, 6, 0, 49, 42, 35, 28, 21, 14, 7, 0, 64, 56, 48, 40, 32, 24, 16, 8, 0, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 0, 121, 110, 99, 88, 77, 66, 55, 44, 33, 22, 11, 0

Offset: 0

Ross La Haye, Mar 02 2004

			Table begins
   0;
   1,  0;
   4,  2,  0;
   9,  6,  3,  0;
  16, 12,  8,  4,  0;
  25, 20, 15, 10,  5,  0;
  36, 30, 24, 18, 12,  6,  0;
  ...
a(5,3) = 40 because 5 * (5 + 3) = 5 * 8 = 40.

Muniru A Asiru, Table of n, a(n) for n = 0..5150 (rows n = 0..100,flattened)
P. De Geest, Palindromic Quasipronics of the form n(n+x)

Columns: a(n, 0) = A000290(n), a(n, 1) = A002378(n), a(n, 2) = A005563(n), a(n, 3) = A028552(n), a(n, 4) = A028347(n+2), a(n, 5) = A028557(n), a(n, 6) = A028560(n), a(n, 7) = A028563(n), a(n, 8) = A028566(n). Diagonals: a(n, n-4) = A054000(n-1), a(n, n-3) = A014107(n), a(n, n-2) = A046092(n-1), a(n, n-1) = A000384(n), a(n, n) = A001105(n), a(n, n+1) = A014105(n), a(n, n+2) = A046092(n), a(n, n+3) = A014106(n), a(n, n+4) = A054000(n+1), a(n, n+5) = A033537(n). Also note that the sums of the antidiagonals = A002411.

Cf. A056536, A056537, A082156.

Flat(List([0..11],j->List([0..j],i->j*(j-i)))); # Muniru A Asiru, Sep 11 2018

Maple
```
seq(seq((j-i)*j,i=0..j),j=0..14);
```

Table[# (# + k) &[m - k], {m, 0, 11}, {k, 0, m}] // Flatten (* Michael De Vlieger, Oct 15 2018 *)

G.f.: x*(1+x-2*x^2*y)/((1-x*y)^2*(1-x)^3). - Vladeta Jovovic, Mar 05 2004

More terms from Emeric Deutsch, Mar 15 2004

A095729 A002260 squared, as an infinite lower triangular matrix, read by rows.

Original entry on oeis.org

1, 3, 4, 6, 10, 9, 10, 18, 21, 16, 15, 28, 36, 36, 25, 21, 40, 54, 60, 55, 36, 28, 54, 75, 88, 90, 78, 49, 36, 70, 99, 120, 130, 126, 105, 64, 45, 88, 126, 156, 175, 180, 168, 136, 81, 55, 108, 156, 196, 225, 240, 238, 216, 171, 100, 66, 130, 189, 240, 280, 306, 315, 304

Offset: 1

Gary W. Adamson, Jun 05 2004, Feb 17 2007

Sum of terms in n-th row = A001296(n-1).
By columns, k; even columns sequences as f(x), x = 1, 2, 3...; = (k/2)x^2 + (k^2 - k/2)x. For example, terms in row 2, (A028552): 4, 10, 18, 28, 40...= x^2 + 3x; row 4 = 2x^2 + 14x, row 6 = 3x^2 + 33x, row 8 = 4x^2 + 60x...etc.
The number in the i-th row and j-th column (j<=i) of the squared matrix is j*(binomial[i + 1, 2] - binomial[j, 2]). - Keith Schneider (schneidk(AT)email.unc.edu), Jul 23 2007

			First few rows of the triangle are
  1;
  3, 4;
  6, 10, 9;
  10, 18, 21, 16;
  15, 28, 36, 36, 25;
  21, 40, 54, 60, 55, 36,
  ...
[1 0 0 / 1 2 0 / 1 2 3]^2 = [1 0 0 / 3 4 0 / 6 10 9].
Next higher order matrix generates rows of the one lower order, plus the next row.
For example, the 4 X 4 matrix [1 0 0 0 / 1 2 0 0 / 1 2 3 0 / 1 2 3 4]^2 = [1 0 0 0 / 3 4 0 0 / 6 10 9 0 / 10 18 21 16].

Cf. A001296, A028552, A002260.

FindRow[n_] := Module[{i = 0}, While[Binomial[i, 2] < n, i++ ]; i - 1]; FindCol[n_] := n - Binomial[FindRow[n], 2]; A095729[n_] := FindCol[n](Binomial[FindRow[n]+1, 2] - Binomial[FindCol[n], 2]); Table[A095729[i], {i, 1, 91}] (* Keith Schneider (schneidk(AT)email.unc.edu), Jul 23 2007 *)

More terms from Keith Schneider (schneidk(AT)email.unc.edu), Jul 23 2007

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A132773 a(n) = n*(n + 31).

Original entry on oeis.org

0, 32, 66, 102, 140, 180, 222, 266, 312, 360, 410, 462, 516, 572, 630, 690, 752, 816, 882, 950, 1020, 1092, 1166, 1242, 1320, 1400, 1482, 1566, 1652, 1740, 1830, 1922, 2016, 2112, 2210, 2310, 2412, 2516, 2622, 2730, 2840, 2952, 3066, 3182, 3300, 3420, 3542, 3666

Offset: 0

Omar E. Pol, Aug 28 2007

Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132758, A132759, A132760.
Cf. A132761, A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769, A132770.
Cf. A132771, A132772.

Table[n*(n + 31), {n, 0, 100}] (* Wesley Ivan Hurt, Apr 16 2024 *)

a(n)=n*(n+31) \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 2*x*(-16+15*x)/(-1+x)^3. - R. J. Mathar, Nov 14 2007
a(n) = 2*A132758(n). - R. J. Mathar, Jul 22 2009
a(n) = 2*n + a(n-1) + 30, with n > 0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(31)/31 = A001008(31)/A102928(31) = 290774257297357/2238255069850800, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/31 - 7313175618421/319750724264400. (End)
From Elmo R. Oliveira, Dec 13 2024: (Start)
E.g.f.: exp(x)*x*(32 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A152919 a(1)=1, for n>1, a(n) = n^2/4 + n/2 for even n, a(n) = n^2/4 + n - 5/4 for odd n.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 18, 20, 28, 30, 40, 42, 54, 56, 70, 72, 88, 90, 108, 110, 130, 132, 154, 156, 180, 182, 208, 210, 238, 240, 270, 272, 304, 306, 340, 342, 378, 380, 418, 420, 460, 462, 504, 506, 550, 552, 598, 600, 648, 650, 700, 702, 754, 756, 810, 812, 868

Offset: 1

Roger L. Bagula, Dec 15 2008

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

a[n_] := If[n == 1, 1, If[Mod[n, 2] == 0, n^2/4 + n/2, n^2/4 + n - 5/4]];
Table[a[n], {n, 1, 100}]

From Chai Wah Wu, Jun 09 2020: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 6.
G.f.: x*(x^5 - x^4 - x - 1)/((x - 1)^3*(x + 1)^2). (End)
From Bernard Schott, Jun 10 2020: (Start)
Bisections are:
a(1) = 1 and a(2k+1) = A028552(k) for k >= 1,
a(2k) = A002378(k) for k >= 1, hence,
a(2k+2) = a(2k+1) + 2 for k >= 1. (End)

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cp's OEIS Frontend

A265611 a(n) = a(n-1) + floor((n-1)/2) - (-1)^n + 2 for n>=2, a(0)=1, a(1)=3.

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A277978 a(n) = 3*n*(n+3).

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A285089 Rectangular array by antidiagonals: row n is the ordered sequence of numbers k that minimize |d(n+1-k) - d(k)|, where d(i) are the divisors of n.

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A367076 Irregular triangle read by rows: T(n,k) (0 <= n, 0 <= k < 2^n). T(n,k) = -Sum_{i=0..k} A365968(n,i).

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A380841 Array read by ascending antidiagonals: A(n,k) = n! * [x^n] 1/(1 - x*exp(x))^k.

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A070895 Triangle read by rows where T(n+1,k)=T(n,k)+n*T(n-1,k) starting with T(n,n)=1 and T(n,k)=0 if n

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Formula

A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

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GAP

Maple

Mathematica

Formula

Extensions

A095729 A002260 squared, as an infinite lower triangular matrix, read by rows.

A277978 a(n) = 3n(n+3).