A005449 -id:A005449 - cp's OEIS frontend

A261780 Number A(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 4, 0, 1, 4, 15, 24, 8, 0, 1, 5, 26, 73, 82, 16, 0, 1, 6, 40, 164, 354, 280, 32, 0, 1, 7, 57, 310, 1031, 1716, 956, 64, 0, 1, 8, 77, 524, 2395, 6480, 8318, 3264, 128, 0, 1, 9, 100, 819, 4803, 18501, 40728, 40320, 11144, 256, 0

Offset: 0

Alois P. Heinz, Aug 31 2015

Also the number of k-compositions of n: matrices with k rows of nonnegative integers with positive column sums and total element sum n.
A(2,2) = 7: (matrices and corresponding marked compositions are given)
  [1 1]  [0 0]  [1 0]  [0 1]  [1]   [2]   [0]
  [0 0]  [1 1]  [0 1]  [1 0]  [1]   [0]   [2]
  1a1a,  1b1b,  1a1b,  1b1a,  2ab,  2aa,  2bb.

			A(3,2) = 24: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a2aa, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1b2bb, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,      1, ...
  0,  1,   2,    3,     4,      5,      6, ...
  0,  2,   7,   15,    26,     40,     57, ...
  0,  4,  24,   73,   164,    310,    524, ...
  0,  8,  82,  354,  1031,   2395,   4803, ...
  0, 16, 280, 1716,  6480,  18501,  44022, ...
  0, 32, 956, 8318, 40728, 142920, 403495, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
E. Grazzini, E. Munarini, M. Poneti, S. Rinaldi, m-compositions and m-partitions: exhaustive generation and Gray code, Pure Math. Appl. 17 (2006), 111-121.
G. Louchard, Matrix Compositions: a Probabilistic analysis, Proc. GASCOM'08, Pure Mathematics and Applications, 19, 2-3, 127-146, 2008.
E. Munarini, M. Poneti, S. Rinaldi, Matrix compositions, Journal of Integer Sequences, Vol. 12 (2009), Article 09.4.8

Columns k=0-10 give: A000007, A011782, A003480, A145839, A145840, A145841, A161434, A261799, A261800, A261801, A261802.
Rows n=0-2 give: A000012, A001477, A005449.
Main diagonal gives A261783.
Cf. A261718 (same for partitions), A261781.

A:= proc(n, k) option remember; `if`(n=0, 1,
      add(A(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[n_, k_] := SeriesCoefficient[(1-x)^k/(2*(1-x)^k-1), {x, 0, n}]; Table[ a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 07 2017 *)

G.f. of column k: (1-x)^k/(2*(1-x)^k-1).
A(n,k) = Sum_{i=0..k} C(k,i) * A261781(n,k-i).
A(n,k) = Sum_{j>=0} (1/2)^(j+1) * binomial(n-1+k*j,n). - Seiichi Manyama, Aug 06 2024

A022289 a(n) = n(31n + 1)/2.

Original entry on oeis.org

0, 16, 63, 141, 250, 390, 561, 763, 996, 1260, 1555, 1881, 2238, 2626, 3045, 3495, 3976, 4488, 5031, 5605, 6210, 6846, 7513, 8211, 8940, 9700, 10491, 11313, 12166, 13050, 13965, 14911, 15888, 16896, 17935

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences of the form n*((2*k+1)*n + 1)/2: A000217 (k=0), A005449 (k=1), A005475 (k=2), A022265 (k=3), A022267 (k=4), A022269 (k=5), A022271 (k=6), A022273 (k=7), A022275 (k=8), A022277 (k=9), A022279 (k=10), A022281 (k=11), A022283 (k=12), A022285 (k=13), A022287 (k=14), this sequence (k=15).

Table[n (31 n + 1)/2, {n, 0, 40}] (* Bruno Berselli, Oct 13 2016 *)
LinearRecurrence[{3,-3,1},{0,16,63},40] (* Harvey P. Dale, Aug 10 2019 *)

a(n)=n*(31*n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 31*n + a(n-1) - 15, for n>0, a(0)=0. - Vincenzo Librandi, Aug 04 2010
G.f.: x*(16 + 15*x)/(1 - x)^3 . - R. J. Mathar, Sep 02 2016
a(n) = A000217(16*n) - A000217(15*n). In general, n*((2*k+1)*n + 1)/2 = A000217((k+1)*n) - A000217(k*n). - Bruno Berselli, Oct 13 2016
E.g.f.: (x/2)*(31*x + 32)*exp(x). - G. C. Greubel, Aug 23 2017

A051867 15-gonal (or pentadecagonal) numbers: n*(13n-11)/2.

Original entry on oeis.org

0, 1, 15, 42, 82, 135, 201, 280, 372, 477, 595, 726, 870, 1027, 1197, 1380, 1576, 1785, 2007, 2242, 2490, 2751, 3025, 3312, 3612, 3925, 4251, 4590, 4942, 5307, 5685, 6076, 6480, 6897, 7327, 7770, 8226, 8695, 9177, 9672, 10180, 10701

Offset: 0

N. J. A. Sloane, Dec 15 1999

Sequence found by reading the line from 0, in the direction 0, 15,... and the parallel line from 1, in the direction 1, 42,..., in the square spiral whose vertices are the generalized 15-gonal numbers. - Omar E. Pol, Jul 18 2012

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Table[n (13n-11)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,1,15},50] (* Harvey P. Dale, Feb 29 2012 *)

a(n)=n*(13*n-11)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(1+12*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011
a(n) = 13*n+a(n-1)-12 (with a(0)=0) - Vincenzo Librandi, Aug 06 2010
a(0)=0, a(1)=1, a(2)=15, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Feb 29 2012
a(13*a(n)+79*n+1) = a(13*a(n)+79*n) + a(13*n+1). - Vladimir Shevelev, Jan 24 2014
Product_{n>=2} (1 - 1/a(n)) = 13/15. - Amiram Eldar, Jan 21 2021
E.g.f.: exp(x)*(x + 13*x^2/2). - Nikolaos Pantelidis, Feb 06 2023
a(n) = A000326(3*n-2) - 7*(n-1)^2. In general, if we let P(k,n) = the n-th k-gonal number, then P(5*k,n) = P(5,k*n-k+1) - A005449(k-1)*(n-1)^2. More generally, if we let SP(k,n) = the n-th second k-gonal number, then for m>2 and k>0, P(m*k,n) = P(m,k*n-k+1) - SP(m,k-1)*(n-1)^2. - Charlie Marion, May 21 2024

A062728 Second 11-gonal (or hendecagonal) numbers: a(n) = n(9n+7)/2.

Original entry on oeis.org

0, 8, 25, 51, 86, 130, 183, 245, 316, 396, 485, 583, 690, 806, 931, 1065, 1208, 1360, 1521, 1691, 1870, 2058, 2255, 2461, 2676, 2900, 3133, 3375, 3626, 3886, 4155, 4433, 4720, 5016, 5321, 5635, 5958, 6290, 6631, 6981, 7340, 7708, 8085, 8471, 8866, 9270

Offset: 0

Floor van Lamoen, Jul 21 2001

Old name: Write 0,1,2,3,4,... in a triangular spiral, then a(n) is the sequence found by reading the line from 0 in the direction 0,8,...

Sequence found by reading the line from 0, in the direction 0, 25, ... and the line from 8, in the direction 8, 51, ..., in the square spiral whose vertices are the generalized 11-gonal numbers A195160. - Omar E. Pol, Jul 24 2012

			The spiral begins:
          15
          / \
        16  14
        /     \
      17   3  13
      /   / \   \
    18   4   2  12
    /   /     \   \
  19   5   0---1  11
  /   /             \
20   6---7---8---9--10

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051682.

Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, A033954, this sequence, A135705.

List([0..50], n-> n*(9*n+7)/2); # G. C. Greubel, May 24 2019

[n*(9*n+7)/2: n in [0..50]]; // G. C. Greubel, May 24 2019

Table[n*(9*n+7)/2, {n,0,50}] (* G. C. Greubel, May 24 2019 *)
LinearRecurrence[{3,-3,1},{0,8,25},50] (* Harvey P. Dale, Sep 06 2019 *)

a(n)=n*(9*n+7)/2 \\ Charles R Greathouse IV, Jun 17 2017

[n*(9*n+7)/2 for n in (0..50)] # G. C. Greubel, May 24 2019

a(n) = n*(9*n+7)/2.
a(n) = 9*n + a(n-1) - 1 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
From Bruno Berselli, Jan 13 2011: (Start)
G.f.: x*(8 + x)/(1 - x)^3.
a(n) = Sum_{i=0..n-1} A017257(i) for n > 0. (End)
a(n) = A218470(9n+7). - Philippe Deléham, Mar 27 2013
E.g.f.: x*(16 + 9*x)*exp(x)/2. - G. C. Greubel, May 24 2019

New name from Bruno Berselli (with the original formula), Jan 13 2011

A140672 a(n) = n(3n + 13)/2.

Original entry on oeis.org

0, 8, 19, 33, 50, 70, 93, 119, 148, 180, 215, 253, 294, 338, 385, 435, 488, 544, 603, 665, 730, 798, 869, 943, 1020, 1100, 1183, 1269, 1358, 1450, 1545, 1643, 1744, 1848, 1955, 2065, 2178, 2294, 2413, 2535, 2660, 2788, 2919, 3053

Offset: 0

Omar E. Pol, May 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

[(3*n^2 + 13*n)/2 : n in [0..80]]; // Wesley Ivan Hurt, Dec 27 2023

Table[n (3 n + 13)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 8, 19}, 50] (* Harvey P. Dale, Dec 16 2011 *)

a(n)=n*(3*n+13)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (3*n^2 + 13*n)/2.
a(n) = 3*n + a(n-1) + 5 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=8, a(2)=19; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Dec 16 2011
G.f.: x*(8 - 5*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (1/2)*(3*x^2 +16*x)*exp(x). - G. C. Greubel, Jul 17 2017

A033568 Second pentagonal numbers with odd index: a(n) = (2n-1)(3*n-1).

Original entry on oeis.org

1, 2, 15, 40, 77, 126, 187, 260, 345, 442, 551, 672, 805, 950, 1107, 1276, 1457, 1650, 1855, 2072, 2301, 2542, 2795, 3060, 3337, 3626, 3927, 4240, 4565, 4902, 5251, 5612, 5985, 6370, 6767, 7176, 7597, 8030, 8475, 8932, 9401, 9882, 10375, 10880, 11397, 11926

Offset: 0

N. J. A. Sloane

Sequence found by reading the segment (1, 2) together with the line (one of the diagonal axes) from 2, in the direction 2, 15, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001318, A005449, A033570, A049452, A049453, A051866.

List([0..50], n-> (2*n-1)*(3*n-1)); # G. C. Greubel, Oct 12 2019

[(2*n-1)*(3*n-1): n in [0..50]]; // G. C. Greubel, Oct 12 2019

seq((2*n-1)*(3*n-1), n=0..50); # G. C. Greubel, Oct 12 2019

Table[(2*n-1)*(3*n-1),{n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 28 2010 *)
LinearRecurrence[{3,-3,1},{1,2,15},50] (* Ray Chandler, Dec 08 2011 *)

a(n)=(2*n-1)*(3*n-1) \\ Charles R Greathouse IV, Sep 24 2015

[(2*n-1)*(3*n-1) for n in range(50)] # G. C. Greubel, Oct 12 2019

G.f.: (1-x+12*x^2)/(1-x)^3.
a(n) = a(n-1) + 12*n - 11 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010
a(n) = 6*n^2 - 5*n + 1 = A051866(n) + 1. - Omar E. Pol, Jul 18 2012
E.g.f.: (1 + x + 6*x^2)*exp(x). - G. C. Greubel, Oct 12 2019
From Amiram Eldar, Feb 18 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + Pi/(2*sqrt(3)) + 2*log(2) - 3*log(3)/2.
Sum_{n>=0} (-1)^n/a(n) = 1 + (1/sqrt(3) - 1/2)*Pi - log(2). (End)

More terms from Ray Chandler, Dec 08 2011

A049777 Triangular array read by rows: T(m,n) = n + n+1 + ... + m = (m+n)(m-n+1)/2.

Original entry on oeis.org

1, 3, 2, 6, 5, 3, 10, 9, 7, 4, 15, 14, 12, 9, 5, 21, 20, 18, 15, 11, 6, 28, 27, 25, 22, 18, 13, 7, 36, 35, 33, 30, 26, 21, 15, 8, 45, 44, 42, 39, 35, 30, 24, 17, 9, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21, 11, 78, 77, 75, 72, 68, 63, 57, 50

Offset: 1

Clark Kimberling

Triangle read by rows, T(n,k) = A000217(n) - A000217(k), 0 <= k < n. - Philippe Deléham, Mar 07 2013
Subtriangle of triangle in A049780. - Philippe Deléham, Mar 07 2013
No primes and all composite numbers (except 2^x) are generated after the first two columns of the square array for this sequence. In other words, no primes and all composites except 2^x are generated when m-n >= 2. - Bob Selcoe, Jun 18 2013
Diagonal sums in the square array equal partial sums of squares (A000330). - Bob Selcoe, Feb 14 2014
From Bob Selcoe, Oct 27 2014: (Start)
The following apply to the triangle as a square array read by rows unless otherwise specified (see Table link);
Conjecture: There is at least one prime in interval [T(n,k), T(n,k+1)]. Since T(n,k+1)/T(n,k) decreases to (k+1)/k as n increases, this is true for k=1 ("Bertrand's Postulate", first proved by P. Chebyshev), k=2 (proved by El Bachraoui) and k=3 (proved by Loo).
Starting with T(1,1), The falling diagonal of the first 2 numbers in each column (read by column) are the generalized pentagonal numbers (A001318). That is, the coefficients of T(1,1), T(2,1), T(2,2), T(3,2), T(3,3), T(4,3), T(4,4) etc. are the generalized pentagonal numbers. These are A000326 and A005449 (Pentagonal and Second pentagonal numbers: n*(3*n+1)/2, respectively), interweaved.
Let D(n,k) denote falling diagonals starting with T(n,k):
Treating n as constant: pentagonal numbers of the form n*k + 3*k*(k-1)/2 are D(n,1); sequences A000326, 005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542 are formed by n = 1 through 12, respectively.
Treating k as constant: D(1,k) are (3*n^2 + (4k-5)*n + (k-1)*(k-2))/2. When k = 2(mod3), D(1,k), is same as D(k+1,1) omitting the first (k-2)/3 numbers in the sequences. So D(1,2) is same as D(3,1); D(1,5) is same as D(6,1) omitting the 6; D(1,8) is same as D(9,1) omitting the 9 and 21; etc.
D(1,3) and D(1,4) are sequences A095794 and A140229, respectively.
(End)

			Rows: {1}; {3,2}; {6,5,3}; ...
Triangle begins:
   1;
   3,  2;
   6,  5,  3;
  10,  9,  7,  4;
  15, 14, 12,  9,  5;
  21, 20, 18, 15, 11,  6;
  28, 27, 25, 22, 18, 13,  7;
  36, 35, 33, 30, 26, 21, 15,  8;
  45, 44, 42, 39, 35, 30, 24, 17,  9;
  55, 54, 52, 49, 45, 40, 34, 27, 19, 10; ...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
M. El Bachraoui, Primes in the interval [2n,3n], Int. J. Contemp. Math. Sciences 1:13 (2006), pp. 617-621.
A. Loo, On the primes in the interval [3n,4n], Int. J. Contemp. Math. Sciences 6 (2011), no. 38, 1871-1882.
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv, arXiv:1212.2785 [math.NT], 2012.

Row sums = A000330.
Cf. A001318 (generalized pentagonal numbers).
Cf. A000217, A002260, A049780, A094728, A095794, A140229.
Cf. A000326, 005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542 (pentagonal numbers of form n*k + 3*k*(k-1)/2).

/* As triangle */ [[(m+n)*(m-n+1) div 2: n in [1..m]]: m in [1.. 15]]; // Vincenzo Librandi, Oct 27 2014

Flatten[Table[(n+k) (n-k+1)/2,{n,15},{k,n}]] (* Harvey P. Dale, Feb 27 2012 *)

{T(n,k) = if( k<1 || nMichael Somos, Oct 06 2007 */

Partial sums of A002260 row terms, starting from the right; e.g., row 3 of A002260 = (1, 2, 3), giving (6, 5, 3). - Gary W. Adamson, Oct 23 2007
Sum_{k=0..n-1} (-1)^k*(2*k+1)*A000203(T(n,k)) = (-1)^(n-1)*A000330(n). - Philippe Deléham, Mar 07 2013
Read as a square array: T(n,k) = k*(k+2n-1)/2. - Bob Selcoe, Oct 27 2014

A110449 Triangle read by rows: T(n,k) = n((2k+1)*n+1)/2, 0<=k<=n.

Original entry on oeis.org

0, 1, 2, 3, 7, 11, 6, 15, 24, 33, 10, 26, 42, 58, 74, 15, 40, 65, 90, 115, 140, 21, 57, 93, 129, 165, 201, 237, 28, 77, 126, 175, 224, 273, 322, 371, 36, 100, 164, 228, 292, 356, 420, 484, 548, 45, 126, 207, 288, 369, 450, 531, 612, 693, 774, 55, 155, 255, 355, 455, 555, 655, 755, 855, 955, 1055

Offset: 0

Reinhard Zumkeller, Jul 21 2005

Row sums give A110450; central terms give A110451;
T(n,0) = A000217(n);
T(n,1) = A005449(n) for n>0;
T(n,2) = A005475(n) for n>1;
T(n,3) = A022265(n) for n>2;
T(n,4) = A022267(n) for n>3;
T(n,5) = A022269(n) for n>4;
T(n,6) = A022271(n) for n>5;
T(n,7) = A022263(n) for n>6;
T(n+1,n-1) = A059270(n) for n>1;
T(n,n-1) = A081436(n) for n>1;
T(n,n) = A085786(n).

			Triangle starts:
0;
1, 2;
3, 7, 11;
6, 15, 24, 33;
10, 26, 42, 58, 74;
...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A126890.

Table[n*((2*k + 1)*n + 1)/2, {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 23 2017 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(n*((2*k+1)*n+1)/2, ", ");); print(););} \\ Michel Marcus, Jun 22 2015

T(n,k) = n*((2*k + 1)*n + 1)/2, 0 <= k <= n.

A140675 a(n) = n(3n + 19)/2.

Original entry on oeis.org

0, 11, 25, 42, 62, 85, 111, 140, 172, 207, 245, 286, 330, 377, 427, 480, 536, 595, 657, 722, 790, 861, 935, 1012, 1092, 1175, 1261, 1350, 1442, 1537, 1635, 1736, 1840, 1947, 2057, 2170, 2286, 2405, 2527, 2652, 2780, 2911, 3045, 3182

Offset: 0

Omar E. Pol, May 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

Table[(n(3n+19))/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,11,25},50] (* Harvey P. Dale, Apr 26 2018 *)

a(n)=n*(3*n+19)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (3*n^2 + 19*n)/2.
a(n) = 3*n + a(n-1) + 8 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: x*(11 - 8*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (1/2)*(3*x^2 + 22*x)*exp(x). - G. C. Greubel, Jul 17 2017

A144074 Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet; 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, 7, 3, 0, 1, 4, 15, 20, 5, 0, 1, 5, 26, 64, 59, 7, 0, 1, 6, 40, 148, 276, 162, 11, 0, 1, 7, 57, 285, 843, 1137, 449, 15, 0, 1, 8, 77, 488, 2020, 4632, 4648, 1200, 22, 0, 1, 9, 100, 770, 4140, 13876, 25124, 18585, 3194, 30, 0, 1, 10, 126

Offset: 0

Alois P. Heinz, Sep 09 2008

Column k > 1 is asymptotic to k^n * exp(2*sqrt(n) - 1/2 + c(k)) / (2 * sqrt(Pi) * n^(3/4)), where c(k) = Sum_{m>=2} 1/(m*(k^(m-1)-1)). - Vaclav Kotesovec, Mar 14 2015

			A(4,1) = 5: {aaaa}, {aaa,a}, {aa,aa}, {aa,a,a}, {a,a,a,a}.
A(2,2) = 7: {aa}, {a,a}, {bb}, {b,b}, {ab}, {ba}, {a,b}.
A(2,3) = 15: {aa}, {a,a}, {bb}, {b,b}, {cc}, {c,c}, {ab}, {ba}, {a,b}, {ac}, {ca}, {a,c}, {bc}, {cb}, {b,c}.
A(3,2) = 20: {aaa}, {a,aa}, {a,a,a}, {bbb}, {b,bb}, {b,b,b}, {aab}, {aba}, {baa}, {a,ab}, {a,ba}, {aa,b}, {a,a,b}, {bba}, {bab}, {abb}, {b,ba}, {b,ab}, {bb,a}, {b,b,a}.
Square array begins:
  1, 1,   1,    1,    1,     1, ...
  0, 1,   2,    3,    4,     5, ...
  0, 2,   7,   15,   26,    40, ...
  0, 3,  20,   64,  148,   285, ...
  0, 5,  59,  276,  843,  2020, ...
  0, 7, 162, 1137, 4632, 13876, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
N. J. A. Sloane, Transforms

Columns k=0-10 give: A000007, A000041, A034899, A144067, A144068, A144069, A144070, A144071, A144072, A144073, A292837.
Rows n=0-2 give: A000012, A001477, A005449.
Main diagonal gives A252654.
Cf. A004248, A257740, A256142, A292804.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->k^j)(n); seq(seq(A(n, d-n), n=0..d), d=0..14);

a[n_, k_] := SeriesCoefficient[ Product[1/(1-x^j)^(k^j), {j, 1, n}], {x, 0, n}]; a[0, ] = 1; a[?Positive, 0] = 0;
Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 15 2014 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d p[d], {d, Divisors[j]}] b[n-j], {j, 1, n}]/n]; b];
A[n_, k_] := etr[k^#&][n];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 30 2020, after Alois P. Heinz *)

G.f. of column k: Product_{j>=1} 1/(1-x^j)^(k^j).
Column k is Euler transform of the powers of k.
T(n,k) = Sum_{i=0..k} C(k,i) * A257740(n,k-i). - Alois P. Heinz, May 08 2015

Name changed by Alois P. Heinz, Sep 21 2018

cp's OEIS Frontend

A261780 Number A(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A022289 a(n) = n*(31*n + 1)/2.

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A051867 15-gonal (or pentadecagonal) numbers: n*(13n-11)/2.

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A062728 Second 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n+7)/2.

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A140672 a(n) = n*(3*n + 13)/2.

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A033568 Second pentagonal numbers with odd index: a(n) = (2*n-1)*(3*n-1).

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A049777 Triangular array read by rows: T(m,n) = n + n+1 + ... + m = (m+n)(m-n+1)/2.

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A022289 a(n) = n(31n + 1)/2.

A062728 Second 11-gonal (or hendecagonal) numbers: a(n) = n(9n+7)/2.

A140672 a(n) = n(3n + 13)/2.

A033568 Second pentagonal numbers with odd index: a(n) = (2n-1)(3*n-1).

A110449 Triangle read by rows: T(n,k) = n((2k+1)*n+1)/2, 0<=k<=n.

A140675 a(n) = n(3n + 19)/2.