A047812 -id:A047812 - cp's OEIS frontend

A007042 Left diagonal of partition triangle A047812.

0, 1, 3, 5, 9, 13, 20, 28, 40, 54, 75, 99, 133, 174, 229, 295, 383, 488, 625, 790, 1000, 1253, 1573, 1956, 2434, 3008, 3716, 4563, 5602, 6840, 8347, 10141, 12308, 14881, 17975, 21635, 26013, 31183, 37336, 44581, 53172, 63259, 75173, 89132, 105556, 124752

Offset: 1

N. J. A. Sloane, R. K. Guy

For n > 2, a(n) is also the number of partitions of n into parts <= n-2: a(n) = A026820(n+1, n-1). - Reinhard Zumkeller, Jan 21 2010
Also, the number of partitions of 2*n in which n-1 is the maximal part; see the Mathematica section. - Clark Kimberling, Mar 13 2012
This is column 2 of the matrix A in Sect. 2.3 of the Govindarajan preprint, cf. references and A096651. - M. F. Hasler, Apr 12 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. Govindarajan, Notes on higher-dimensional partitions, arXiv:1203.4419 [math.CO], 2012.
R. K. Guy, Letter to N. J. A. Sloane, Aug. 1992.
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Preprint, 1992. (Annotated scanned copy)
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Cf. A000041, A007044, A007045, A026820, A047812, A051643, A096651.

Column k = 2 of A081719.

using Nemo
function A007042List(len)
    R, z = PolynomialRing(ZZ, "z")
    e = eta_qexp(-1, len+2, z)
    [coeff(e, j) - 2 for j in 2:len+1] end
A007042List(45) |> println # Peter Luschny, May 30 2020

f[n_]:= Length[Select[IntegerPartitions[2 n], First[#]==n-1 &]]; Table[f[n], {n, 1, 24}] (* Clark Kimberling, Mar 13 2012 *)
a[n_]:= PartitionsP[n+1]-2; Table[a[n], {n,1,50}] (* Jean-François Alcover, Jan 28 2015, after M. F. Hasler *)

A007042(n)=numbpart(n+1)-2  \\ M. F. Hasler, Apr 12 2012

a(n) = A000041(n+1) - 2. - Vladeta Jovovic, Oct 06 2001

More terms from James Sellers

Name edited by Petros Hadjicostas, May 31 2020

A007044 Left diagonal of partition triangle A047812.

Original entry on oeis.org

0, 0, 1, 7, 20, 48, 100, 194, 352, 615, 1034, 1693, 2705, 4239, 6522, 9889, 14786, 21844, 31913, 46165, 66162, 94035, 132600, 185637, 258128, 356674, 489906, 669173, 909212, 1229217, 1653993, 2215597, 2955192, 3925659, 5194520, 6847963, 8995524, 11776227

Offset: 1

N. J. A. Sloane and R. K. Guy

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..500
R. K. Guy, Letter to N. J. A. Sloane, Aug. 1992
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Preprint, 1992. (Annotated scanned copy)
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Cf. A007042, A007045, A047812, A051643.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<0
      or t*i b(2*n+2, n$2):
seq(a(n), n=1..50);  # Alois P. Heinz, May 31 2020

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[n < 0 || t i < n, 0, b[n, i - 1, t] + b[n - i, Min[i, n - i], t - 1]]];
a[n_] := b[2n+2, n, n];
Array[a, 50] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

T(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
for(n=1, 40, print1(T(n, 2), ",")) \\ Petros Hadjicostas, May 31 2020

Name edited by Petros Hadjicostas, May 31 2020

A007045 Second (lower) diagonal of partition triangle A047812.

Original entry on oeis.org

0, 1, 5, 20, 51, 112, 221, 411, 720, 1221, 2003, 3206, 5021, 7728, 11698, 17472, 25766, 37580, 54254, 77617, 110087, 154942, 216488, 300456, 414365, 568113, 774571, 1050572, 1417868, 1904641, 2547152, 3392042, 4498948, 5944158, 7824703, 10263932, 13418043, 17484554

Offset: 2

N. J. A. Sloane and R. K. Guy

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 2..500 (terms n=2..53 from Vincenzo Librandi, n=54..103 from Bruno Berselli)
R. K. Guy, Letter to N. J. A. Sloane, Aug. 1992.
R. K. Guy, Parker's permutation problem involves the Catalan numbers, preprint, 1992. (Annotated scanned copy)
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Cf. A007042, A007044, A047812, A051643.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<0
      or t*i b((n-3)*(n+1), n$2):
seq(a(n), n=2..40);  # Alois P. Heinz, May 31 2020

s[n_] := s[n] = Series[Product[(1 - q^(2*n - k))/(1 - q^(k + 1)), {k, 0, n - 1}], {q, 0, n^2}]; t[n_, k_] := SeriesCoefficient[s[n], k*(n + 1)]; A007045 = Join[{0}, Table[t[n + 3, n], {n, 0, 25}] ] (* Jean-François Alcover, Apr 25 2012 *)

T(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
for(n=3, 33, print1(T(n, n-3), ", ")) \\ Petros Hadjicostas, May 31 2020

A136621 Transpose T(n,k) of Parker's partition triangle A047812 (n >= 1 and 0 <= k <= n-1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 5, 1, 1, 11, 20, 9, 1, 1, 18, 51, 48, 13, 1, 1, 26, 112, 169, 100, 20, 1, 1, 38, 221, 486, 461, 194, 28, 1, 1, 52, 411, 1210, 1667, 1128, 352, 40, 1, 1, 73, 720, 2761, 5095, 4959, 2517, 615, 54, 1, 1, 97, 1221, 5850, 13894, 18084, 13241, 5288, 1034, 75, 1

Offset: 1

Alford Arnold, Jan 26 2008

Parker's triangle is closely associated with q-binomial coefficients and Gaussian polynomials; cf. A063746. For example, row 4 of A063746 is 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, the coefficients of [8, 4], while the entries in row 4 of A047812 are the coefficients of q^(k*(4+1)) = q^(5*k) in [8, 4] where k runs from 0 to n-1 = 3. Likewise, by symmetry, "1 7 5 1" is embedded also because they are the coefficients of q^(5*(3-k)), where k runs from 0 to n-1 = 3. [Edited by Petros Hadjicostas, May 30 2020]

			Row four of A047812 is 1 5 7 1, so row four of the present entry is 1 7 5 1.
From _Petros Hadjicostas_, May 30 2020: (Start)
Triangle T(n,k) (with rows n >= 1 and columns k = 0..n-1) begins:
  1;
  1,  1;
  1,  3,   1;
  1,  7,   5,    1;
  1, 11,  20,    9,    1;
  1, 18,  51,   48,   13,    1;
  1, 26, 112,  169,  100,   20,   1;
  1, 38, 221,  486,  461,  194,  28,  1;
  1, 52, 411, 1210, 1667, 1128, 352, 40, 1;
  ... (End)

Alois P. Heinz, Rows n = 1..141, flattened
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.
Wikipedia, E. T. Parker.

Cf. A000108 (Catalan row sums), A047812, A063746.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i
       b((n-k-1)*(n+1), n$2):
seq(seq(T(n, k), k=0..n-1), n=1..12);  # Alois P. Heinz, May 30 2020

T[n_, k_]:= SeriesCoefficient[QBinomial[2*n, n, q], {q, 0, k*(n+1)}];
Table[T[n, n-k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, May 31 2020 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[n < 0 || t i < n, 0, b[n, i - 1, t] + b[n - i, Min[i, n - i], t - 1]]];
T[n_, k_] := b[(n-k-1)(n+1), n, n];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

T(n, k) = #partitions(k*(n+1), n, n);
for (n=1, 10, for (k=0, n-1, print1(T(n, n-1-k), ", "); ); print(); ); \\ Petros Hadjicostas, May 30 2020
/* Second program, courtesy of G. C. Greubel */
T(n,k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
vector(12, n, vector(n, k, T(n,n-k))) \\ Petros Hadjicostas, May 31 2020

def T(n,k):
    P. = PowerSeriesRing(ZZ, k*(n+1)+1)
    return P( q_binomial(2*n, n, x) ).list()[k*(n+1)]
[[ T(n,n-k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 31 2020

Name edited by Petros Hadjicostas, May 30 2020

A128567 Matrix square, T(n,k), of Parker's partition triangle A047812, read by rows (n >= 1 and 0 <= k <= n-1).

Original entry on oeis.org

1, 2, 1, 5, 6, 1, 14, 31, 14, 1, 42, 133, 117, 22, 1, 132, 587, 813, 300, 36, 1, 429, 2531, 4871, 2896, 692, 52, 1, 1430, 10950, 27743, 23961, 9206, 1430, 76, 1, 4862, 47185, 151208, 175734, 96418, 24598, 2798, 104, 1, 16796, 203704, 804065, 1200301, 882471, 329426, 62885, 5236, 146, 1

Offset: 1

Paul D. Hanna, Mar 12 2007

Column 0 is the Catalan numbers (A000108). Parker's partition triangle may be defined as: A047812(n,k) = [q^(n*k+k)] in the central q-binomial coefficient [2*n,n] for n >= 1 and 0 <= k <= n-1. [Edited by Petros Hadjicostas, May 30 2020]

			Triangle T(n,k) (with rows n >= 1 and columns k = 0..n-1) begins:
      1;
      2,      1;
      5,      6,      1;
     14,     31,     14,       1;
     42,    133,    117,      22,      1;
    132,    587,    813,     300,     36,      1;
    429,   2531,   4871,    2896,    692,     52,     1;
   1430,  10950,  27743,   23961,   9206,   1430,    76,    1;
   4862,  47185, 151208,  175734,  96418,  24598,  2798,  104,   1;
  16796, 203704, 804065, 1200301, 882471, 329426, 62885, 5236, 146, 1;
  ...

R. K. Guy, Parker's permutation problem involves the Catalan numbers, preprint, 1992. (Annotated scanned copy)
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.
Wikipedia, E. T. Parker.

Cf. A000108 (column k=0), A047812, A128568 (column k=1), A128569 (column k=2), A128602 (row sums).

{T(n, k)=local(M);M=matrix(n+1,n+1,r,c,if(rPetros Hadjicostas, May 31 2020

T(n,k) = Sum_{s=k..n-1} A047812(n,s)*A047812(s+1,k) for n >= 1 and 0 <= k <= n-1. - Petros Hadjicostas, May 31 2020

Name edited and offset changed by Petros Hadjicostas, May 30 2020

A137614 Triangle read by rows: A000012 * A047812 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 18, 28, 12, 1, 6, 31, 76, 63, 19, 1, 7, 51, 176, 232, 131, 27, 1, 8, 79, 370, 693, 617, 248, 39, 1, 9, 119, 722, 1821, 2284, 1458, 450, 53, 1, 10, 173, 1337, 4338, 7243, 6553, 3211, 773, 74, 1

Offset: 0

Gary W. Adamson, Jan 30 2008

Row sums = A014138: (1, 3, 8, 22, 64, 196, 625, ...).
From Petros Hadjicostas, Jun 01 2020: (Start)
We prove the claim above. From Guy (1992, 1993), we know that A000108(n) = Sum_{k=0..n-1} A047812(k) (the row sums of Parker's triangle are Catalan numbers).
We then have Sum_{k=0..n-1} T(n,k) = Sum_{k=0..n-1} Sum_{s=k+1..n} A047812(s,k) = Sum_{s=1..n} Sum_{k=0..s-1} A047812(s,k) = Sum_{s=1..n} A000108(s) = A014138(n) because A014138 contains partial sums of the Catalan numbers. (End)

			Triangle T(n,k) (with rows n >= 1 and columns k = 0..n-1) begins:
  1;
  2,  1;
  3,  4,   1;
  4,  9,   8,   1;
  5, 18,  28,  12,   1;
  6, 31,  76,  63,  19,  1;
  7, 51, 176, 232, 131, 27, 1;
  ...

R. K. Guy, Parker's permutation problem involves the Catalan numbers, preprint, 1992. (Annotated scanned copy)
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Cf. A000012, A014138, A047812.

A(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
T(n,k) = sum(s=k+1, n, A(s,k));
vector(15, n, vector(n, k, T(n, k-1))) \\ Petros Hadjicostas, Jun 01 2020

T(n,k) = Sum_{s=k+1..n} A047812(s,k) for n >= 1 and 0 <= k <= n-1. - Petros Hadjicostas, Jun 01 2020

A335323 First lower diagonal of Parker's triangle A047812.

Original entry on oeis.org

0, 1, 3, 7, 11, 18, 26, 38, 52, 73, 97, 131, 172, 227, 293, 381, 486, 623, 788, 998, 1251, 1571, 1954, 2432, 3006, 3714, 4561, 5600, 6838, 8345, 10139, 12306, 14879, 17973, 21633, 26011, 31181, 37334, 44579, 53170, 63257, 75171, 89130, 105554, 124750, 147269

Offset: 1

Petros Hadjicostas, May 31 2020

nonn

Apparently, this sequence was originally intended to be A7043 (now A007043), but for some reason it was crossed out on p. 4 of the annotated copy of Guy's 1992 preprint.

a(n) is the number of partitions of (n-2)*(n+1) into at most n parts each no bigger than n. Thus, a(n) is the coefficient of q^((n-2)*(n+1)) in the q-binomial coefficient [2*n, n].

			a(1) = 0 because it does not make sense to talk about the partitions of (1-2)*(1+1) = -2.
a(2) = 1 because we have only the empty partition for (2-2)*(2+1) = 0.
a(3) = 3 because we have the following partitions of (3-2)*(3+1) = 4 into no more than 3 parts each no bigger than 3: 1+3 = 1+1+2 = 2+2.
a(4) = 7 because we have the following partitions of (4-2)*(4+1) = 10 into no more than 4 parts each no bigger than 4: 2+4+4 = 3+3+4 = 1+1+4+4 = 1+2+3+4 = 1+3+3+3 = 2+2+2+4 = 2+2+3+3.
The PARI function partitions((n-2)*(n+1), n, n) can generate these partitions.

Alois P. Heinz, Table of n, a(n) for n = 1..500
R. K. Guy, Letter to N. J. A. Sloane, Aug. 1992.
R. K. Guy, Parker's permutation problem involves the Catalan numbers, preprint, 1992. (Annotated scanned copy)
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Cf. A007042, A007043, A007044, A007045, A047812, A051643.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<0
      or t*i b((n-2)*(n+1), n$2):
seq(a(n), n=1..50);  # Alois P. Heinz, May 31 2020

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[n < 0 || t i < n, 0, b[n, i - 1, t] + b[n - i, Min[i, n - i], t - 1]]];
a[n_] := b[(n-2)(n+1), n, n];
Array[a, 50] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

T(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
for(n=1, 43, print1(T(n, n-2), ", "))

A128545 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk) in the q-binomial coefficient [2n, n] for n >= k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 8, 5, 1, 1, 7, 18, 18, 7, 1, 1, 11, 39, 58, 39, 11, 1, 1, 15, 75, 155, 155, 75, 15, 1, 1, 22, 141, 383, 526, 383, 141, 22, 1, 1, 30, 251, 867, 1555, 1555, 867, 251, 30, 1, 1, 42, 433, 1860, 4192, 5448, 4192, 1860, 433, 42, 1

Offset: 0

Paul D. Hanna, Mar 10 2007

Variant of A047812 (Parker's partition triangle).

Column 1 equals the number of partitions of n: A000041(n) is the coefficient of q^n in the central q-binomial coefficient [2*n, n] for n > 0.

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,    1;
  1,  5,   8,    5,    1;
  1,  7,  18,   18,    7,    1;
  1, 11,  39,   58,   39,   11,    1;
  1, 15,  75,  155,  155,   75,   15,    1;
  1, 22, 141,  383,  526,  383,  141,   22,   1;
  1, 30, 251,  867, 1555, 1555,  867,  251,  30,  1;
  1, 42, 433, 1860, 4192, 5448, 4192, 1860, 433, 42, 1;
  ...

Paul D. Hanna, Rows n = 0..45, flattened.

Cf. A003239, A047812 (variant), A047996, A123610, A123611 (row sums).

Cf. A000041 (column 1), A128552 (column 2), A128553 (column 3), A128554 (column 4).

PARI
```
T(n,k)=if(n
				
```

Row sums equal the row sums of triangle A123610: A123611(n) = 2*A047996(2*n,n) = 2*A003239(n) for n > 0, where A047996 is the triangle of circular binomial coefficients and A003239(n) = number of rooted planar trees with n non-root nodes.

A051643 Central elements in Parker's partition triangle.

Original entry on oeis.org

1, 3, 20, 169, 1667, 18084, 208960, 2527074, 31630390, 406680465, 5342750699, 71442850111, 969548468960, 13323571588607, 185072895183632, 2594890728951909, 36681505784903758, 522291180086851188, 7484621370716999785, 107876522368295972285, 1562916545414144667559

Offset: 0

James Sellers

Alois P. Heinz, Table of n, a(n) for n = 0..90
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Cf. A007042, A047812, A136621.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i
       b(2*n*(n+1), 2*n+1$2):
seq(a(n), n=0..20);  # Alois P. Heinz, May 30 2020

a[n_] := SeriesCoefficient[QBinomial[2(2n+1), 2n+1, q], {q, 0, 2n(n+1)}];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 19 2019 *)

a(n) = coefficient of q^((m^2-1)/2) = q(2*n*(n+1)) in the q-binomial coefficient [2*m, m] = [2*(2*n+1), 2*n+1], where m = 2*n+1. [Corrected by Petros Hadjicostas, May 30 2020]

a(n) is the number of partitions of 2*n*(n+1) into at most 2*n+1 parts each no bigger than 2*n+1. - Petros Hadjicostas, May 30 2020

a(18)-a(20) from Alois P. Heinz, May 30 2020

A081719 Triangle T(n,k) read by rows, related to Faà di Bruno's formula (n >= 0 and 0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 5, 1, 0, 1, 9, 14, 7, 1, 0, 1, 13, 32, 27, 9, 1, 0, 1, 20, 66, 80, 44, 11, 1, 0, 1, 28, 123, 203, 160, 65, 13, 1, 0, 1, 40, 222, 465, 486, 280, 90, 15, 1, 0, 1, 54, 377, 985, 1305, 990, 448, 119, 17, 1, 0, 1, 75, 630, 1978, 3203, 3051, 1807, 672, 152, 19, 1

Offset: 0

N. J. A. Sloane, Apr 05 2003

From Petros Hadjicostas, May 30 2020: (Start)
We may prove Philippe Deléham's formula by induction on n. Let P(n,k) = A008284(n,k) and b(n) = A039809(n). For n = 0, Sum_{k=0..0} T(0,k) = 1 = b(1). Let n >= 1, and assume his formula is true for all s < n, i.e., Sum_{k=0..s} T(s,k) = b(s+1).
Then Sum_{k=0..n} T(n, k) = Sum_{k=1..n} T(n,k) = Sum_{k=1..n} Sum_{s=k-1..n-1} P(n+1, s+1)*T(s, k-1) = Sum_{s=0..n-1} P(n+1, s+1) Sum_{k=1..s+1} T(s, k-1) = Sum_{s=0..n-1} P(n+1, s+1) Sum_{m=0..s} T(s,m) = Sum_{s=0..n-1} P(n+1, s+1)*b(s+1) = Sum_{r=1..n} P(n+1, r)*b(r) = b(n+1) (by the definition of b = A039809). (End)

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  3,  1;
  0, 1,  5,  5,  1;
  0, 1,  9, 14,  7,  1;
  0, 1, 13, 32, 27,  9,  1;
  0, 1, 20, 66, 80, 44, 11, 1;
  ...

Warren P. Johnson, The curious history of Faà di Bruno's formula, American Mathematical Monthly, 109 (2002), 217-234.
Warren P. Johnson, The curious history of Faà di Bruno's formula, American Mathematical Monthly, 109 (2002), 217-234.
Winston C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Cf. A007042, A008284, A039809, A047812.

(* b = A008284 *)
b[n_, k_]:= b[n, k]= If[n>0 && k>0, b[n-1, k-1] + b[n-k, k], Boole[n==0 && k==0]];
T[n_, k_]:= T[n, k]= If[n==0 && k==0, 1, If[k==0, 0,  Sum[T[j, k-1]*b[n+1, j+1], {j, k-1, n-1}] ]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 31 2020 *)

P(n, k)=#partitions(n-k, k); /* A008284 */
tabl(nn) = {A = matrix(nn, nn, n, k, 0); A[1,1] = 1; for(n=2, nn, for(k=2, n, A[n,k] = sum(s=k-2, n-2, P(n, s+1)*A[s+1,k-1])));
for (n=1, nn, for (k=1, n, print1(A[n, k], ", "); ); print(); ); }  \\ Petros Hadjicostas, May 29 2020

There is a recurrence involving the partition function A008284.
Sum_{k=0..n} T(n,k) = A039809(n+1). - Philippe Deléham, Sep 30 2006
From Petros Hadjicostas, May 30 2020: (Start)
T(n, k) = Sum_{s=k-1..n-1} A008284(n+1, s+1)*T(s, k-1) for 1 <= k <= n with T(0,0) = 1 and T(n,0) = 0 for n >= 1.
T(n, k=2) = A007042(n) = A047812(n,2). (End)

More terms from Emeric Deutsch, Feb 28 2005

cp's OEIS Frontend

A007042 Left diagonal of partition triangle A047812.

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A007044 Left diagonal of partition triangle A047812.

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A007045 Second (lower) diagonal of partition triangle A047812.

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A136621 Transpose T(n,k) of Parker's partition triangle A047812 (n >= 1 and 0 <= k <= n-1).

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A128567 Matrix square, T(n,k), of Parker's partition triangle A047812, read by rows (n >= 1 and 0 <= k <= n-1).

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A137614 Triangle read by rows: A000012 * A047812 as infinite lower triangular matrices.

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A335323 First lower diagonal of Parker's triangle A047812.

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A128545 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk) in the q-binomial coefficient [2n, n] for n >= k >= 0.