A026820 -id:A026820 - cp's OEIS frontend

A117488 The number 1 followed by 2*k+1 terms from column k of table A115994.

1, 1, 2, 3, 1, 2, 5, 8, 14, 1, 2, 5, 10, 18, 30, 49, 1, 2, 5, 10, 20, 34, 59, 94, 149, 1, 2, 5, 10, 20, 36, 63, 104, 169, 264, 405, 1, 2, 5, 10, 20, 36, 65, 108, 179, 284, 445, 676, 1017, 1, 2, 5, 10, 20, 36, 65, 110, 183, 294, 465, 716, 1089, 1622, 2387, 1, 2, 5, 10, 20, 36, 65

Offset: 1

Alford Arnold, Mar 22 2006

			Column two of table A115994 begins 1 2 5 8 14 20 30 ... A006918
so row three of A117488 is 1 2 5 8 14.
Triangle starts
1
1 2 3
1 2 5 8 14
1 2 5 10 18 30 49
1 2 5 10 20 34 59 94 149
1 2 5 10 20 36 63 104 169 264 405
1 2 5 10 20 36 65 108 179 284 445 676 1017
1 2 5 10 20 36 65 110 183 294 465 716 1089 1622 2387

Cf. A006918, A115994, A117489.

A026820 := proc(n,k) if k > n then combinat[numbpart](n,n) ; else combinat[numbpart](n,k) ; fi ; end: A115994 := proc(n,k) local i ; add(A026820(i,k)*A026820(n-k^2-i,k),i=0..n-k^2) ; end: A117488 := proc(n,k) if k >= 2*n then 0 ; else if n = 1 then 1; else A115994(k+n^2-2*n,n-1) ; fi ; fi ; end: for n from 1 to 10 do for k from 1 to 2*n-1 do printf("%d ",A117488(n,k)) ; od ; od ; # R. J. Mathar, Feb 22 2007

More terms from R. J. Mathar, Feb 22 2007

A097306 Array of number of partitions of n with odd parts not exceeding 2*m-1 with m in {1, 2, ..., ceiling(n/2)}.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 3, 4, 1, 3, 4, 5, 1, 3, 5, 6, 1, 4, 6, 7, 8, 1, 4, 7, 9, 10, 1, 4, 8, 10, 11, 12, 1, 5, 9, 12, 14, 15, 1, 5, 10, 14, 16, 17, 18, 1, 5, 11, 16, 19, 21, 22, 1, 6, 13, 19, 23, 25, 26, 27, 1, 6, 14, 21, 26, 29, 31, 32, 1, 6, 15, 24, 30, 34, 36, 37, 38, 1, 7, 17, 27

Offset: 1

Wolfdieter Lang, Aug 13 2004

The sequence of row lengths of this array is A008619 = [1,1,2,2,3,3,4,4,5,5,6,6,7,7,...].
This is the partial row sums array of array A097305.
The number of partitions of N=2*n (n >= 1) into even parts not exceeding 2*k, with k from {1,...,n}, is given by the triangle A026820(n,k).

			[1]; [1]; [1,2]; [1,2]; [1,2,3]; [1,3,4]; [1,3,4,5]; [1,3,5,6]; ...
T(8,2)=3 because there are three partitions of 8 with odd parts not exceeding 3, namely (1^8), (1^5,3) and (1^2,3^2).
T(6,2)=3 from the partitions (1^6), (1^3,3) and (3^2).

Wolfdieter Lang, First 18 rows.

Row sums: A097307.

Sequence of row numbers for n>=1: [seq(coeff(series(product(1/(1-x^(2*k-1)),k=1..p),x,n+1),x,n),p=1..ceil(n/2))].

T(n, m) = number of partitions of n with odd parts only and largest parts <= 2*m-1 for m from {1, 2, ..., ceiling(n/2)}.

T(n, m) = Sum_{k=1..m} A097305(n, k), m = 1..ceiling(n/2), n >= 1.

A249770 Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order n with k invariant factors (2 <= n, 1 <= k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Álvar Ibeas, Nov 06 2014

The length of n-th row is A051903(n) and its last element is A249773(A101296(n)).

T(n,k) depends only on k and the prime signature of n.

			First rows:
1;
1;
1,1;
1;
1;
1;
1,1,1;
1,1;
1;
1;
1,1;
1;
1;
1;
1,2,1,1;
1;
...

Álvar Ibeas, Rows n=2..58654, flattened

Refinement of A000688.

Cf. A008284, A051903, A026820, A101296, A249771, A249773, A046951, A264809.

f[{x_, y_}] := x^IntegerPartitions[y];
g[n_] := FactorInteger[n][[1, 1]];
h[list_] := Apply[Times,Map[PadRight[#, Max[Map[Length, SplitBy[list, g]]], 1] &,SplitBy[list, g]]]; t[list_] := Tally[Map[Length, list]][[All, 2]];
Map[t, Table[Map[h, Join @@@ Tuples[Map[f, FactorInteger[n]]]], {n, 2, 50}]] // Grid (* Geoffrey Critzer, Nov 26 2015 *)

T(n,k) = A249771(A101296(n),k).
T(n,1) = 1. If k > 1 and n = Product(p_i^e_i), T(n,k) = Sum(Product(A008284(e_i,k), i in I) * Product(A026820(e_i,k-1), i not in I)), where the sum is taken over nonempty subsets I of {1,...,omega(n)}.
If p is prime and gcd(p,n) = 1, T(pn,k) = T(n,k).
Dirichlet g.f. of column sums: zeta(s)zeta(2s)···zeta(ms) = 1 + Sum_{n >= 2} (Sum_{k=1..m} T(n,k)) / n^s.
T(n,1) + T(n,2) = A046951(n)

A249771 Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order A025487(n) with k invariant factors (2 <= n, 1 <= k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 5, 2, 2, 1, 3, 1, 3, 3, 2, 1, 1, 1, 1, 3, 5, 1, 2

Offset: 2

Álvar Ibeas, Nov 06 2014

The length of n-th row is A051282(n).

Signatures differing only by a (trailing) list of ones give identical rows.

			First rows:
1;
1,1;
1;
1,1,1;
1,1;
1,2,1,1;
1,1,1;
1;
1,2,2,1,1;
1,3;
...

Álvar Ibeas, Rows n=2..1075, flattened

Refinement of A050360. Last row elements: A249773. Cf. A249770, A052304.

T(n,1) = 1. If k > 1 and the prime signature is (e_1,...,e_s), T(n,k) = Sum(Product(A008284(e_i,k), i in I) * Product(A026820(e_i,k-1), i not in I)), where the sum is taken over nonempty subsets I of {1,...,s}.
T(n,k) = A249770(A025487(n),k).
T(n,1) + T(n,2) = A052304(n).

A008640 Number of partitions of n into at most 11 parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 99, 131, 169, 219, 278, 355, 445, 560, 695, 863, 1060, 1303, 1586, 1930, 2331, 2812, 3370, 4035, 4802, 5708, 6751, 7972, 9373, 11004, 12866, 15021, 17475, 20298, 23501, 27169, 31316, 36043, 41373, 47420, 54218, 61903, 70515, 80215, 91058, 103226, 116792, 131970, 148847

Offset: 0

N. J. A. Sloane

For n>10: also number of partitions of n into parts <= 11: a(n)=A026820(n,11). [Reinhard Zumkeller, Jan 21 2010]

A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.

T. D. Noe, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 360
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, -1, -1, -1, -2, -1, -1, 0, -1, 2, 2, 2, 2, 1, 1, 0, -1, -1, -2, -2, -2, -2, 1, 0, 1, 1, 2, 1, 1, 1, 0, 0, -1, -2, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, -1, -1, 1).

Differs from A008634 at 55th term.

a(n) = A008284(n+11, 11), n >= 0.

1/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/(1-x^5)/(1-x^6)/(1-x^7)/(1-x^8)/(1-x^9)/(1-x^10)/(1-x^11)
with(combstruct):ZL12:=[S,{S=Set(Cycle(Z,card<12))},unlabeled]: seq(count(ZL12,size=n),n=0..44); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=11)},unlabelled]: seq(combstruct[count](B, size=n), n=0..44); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 11} ], {x, 0, 60} ], x ]

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-14) + 2*a(n-15) + a(n-16) - a(n-19) - a(n-20) - a(n-21) - 2*a(n-22) - a(n-23) - a(n-24) - a(n-26) + 2*a(n-27) + 2*a(n-28) + 2*a(n-29) + 2*a(n-30) + a(n-31) + a(n-32) - a(n-34) - a(n-35) - 2*a(n-36) - 2*a(n-37) - 2*a(n-38) - 2*a(n-39) + a(n-40) + a(n-42) + a(n-43) + 2*a(n-44) + a(n-45) + a(n-46) + a(n-47) - a(n-50) - 2*a(n-51) - a(n-52) + a(n-59) + a(n-61) - a(n-64) - a(n-65) + a(n-66). - David Neil McGrath, Jul 27 2015

G.f.: 1 / prod(k=1..11, 1 - x^k ). - Joerg Arndt, Aug 04 2015

A008641 Number of partitions of n into at most 12 parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 100, 133, 172, 224, 285, 366, 460, 582, 725, 905, 1116, 1380, 1686, 2063, 2503, 3036, 3655, 4401, 5262, 6290, 7476, 8877, 10489, 12384, 14552, 17084, 19978, 23334, 27156, 31570, 36578, 42333, 48849, 56297

Offset: 0

N. J. A. Sloane

With a different offset, number of partitions of n in which the greatest part is 12.

Also number of partitions of n into parts <= 12: a(n)=A026820(n,12). [Reinhard Zumkeller, Jan 21 2010]

A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.

T. D. Noe, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 361
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, -1, 0, 2, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, -2, 0, 1, 2, 2, 2, 2, 1, 1, 0, -1, -2, -1, -4, -1, -2, -1, 0, 1, 1, 2, 2, 2, 2, 1, 0, -2, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 2, 0, -1, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, -1).

a(n) = A008284(n+12, 12), n >= 0.

Cf. A026810, A026811, A026812, A026813, A026814, A026815, A026816.

1/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/(1-x^5)/(1-x^6)/(1-x^7)/(1-x^8)/(1-x^9)/(1-x^10)/(1-x^11)/(1-x^12)
with(combstruct):ZL13:=[S,{S=Set(Cycle(Z,card<13))}, unlabeled]:seq(count(ZL13,size=n),n=0..46); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=12)},unlabelled]: seq(combstruct[count](B, size=n), n=0..46); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 12} ], {x, 0, 60} ], x ]
Table[ Length[ Select[ Partitions[n], First[ # ] == 12 & ]], {n, 1, 60} ]

G.f.: 1/Product_{k=1..12}(1-x^k).

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) - a(n-13) + 2*a(n-15) + a(n-16) + a(n-17) - a(n-20) - a(n-21) - 2*a(n-22) - a(n-23) - a(n-24) - 2*a(n-26) + a(n-28) + 2*a(n-29) + 2*a(n-30) + 2*a(n-31) + 2*a(n-32) + a(n-33) + a(n-34) - a(n-36) - 2*a(n-37) - a(n-38) - 4*a(n-39) - a(n-40) - 2*a(n-41) - a(n-42) + a(n-44) + a(n-45) + 2*a(n-46) + 2a(n-47) + 2*a(n-48) + 2*a(n-49) + a(n-50) - 2*a(n-52) - a(n-54) - a(n-55) - 2*a(n-56) - a(n-57) - a(n-58) + a(n-61) + a(n-62) + 2*a(n-63) - a(n-65) + a(n-66) - a(n-71) - a(n-73) + a(n-76) + a(n-77) - a(n-78). - David Neil McGrath, Jul 28 2015

More terms from Robert G. Wilson v, Dec 11 2000

A134737 Number of partitions of the n-th partition number into positive parts not greater than n.

Original entry on oeis.org

1, 2, 3, 6, 13, 44, 131, 638, 3060, 22367, 167672, 2127747, 26391031, 537973241, 12274276512, 429819314124, 16928838590640, 1068323095351171, 75345432929798690, 8339062208354516217, 1083103359596125913021, 209256696715820656730807, 48414226122932084106352434

Offset: 1

Reinhard Zumkeller, Nov 07 2007

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..55
Eric Weisstein's World of Mathematics, Partition
Eric Weisstein's World of Mathematics, Partition Function P
Index entries for sequences related to partitions

with(numtheory): P:= proc(n) local d, j; P(n):= `if`(n=0, 1, add(add(d, d=divisors(j)) *P(n-j), j=1..n)/n) end: b:= proc(n,i) if n<0 then 0 elif n=0 then 1 elif i=0 then 0 else b(n,i):= b(n, i-1) +b(n-i, i) fi end: a:= n-> b(P(n),n): seq(a(n), n=1..25); # Alois P. Heinz, Jul 17 2009

(* first do *) Needs["DiscreteMath`IntegerPartitions`"] (* then *) a[n_] := Length@ IntegerPartitions[ PartitionsP[n], n] (* Robert G. Wilson v, Nov 11 2007 *)
P[n_] := P[n] = Module[{d, j}, If[n == 0, 1, Sum[DivisorSum[j, #&]*P[n - j], {j, 1, n}]/n]]; b [n_, i_] := b[n, i] = Which[n<0, 0, n == 0, 1, i == 0, 0, True, b[n, i] = b[n, i-1] + b[n-i, i]]; a[n_] := b[P[n], n]; Table [a[n], {n, 1, 25}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) = A026820(A026820(n,n),n) = A026820(A000041(n),n).

More terms from Alois P. Heinz, Jul 17 2009

A215520 Number T(n,k) of distinct values of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n with largest part <= k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 4, 7, 9, 10, 11, 1, 4, 8, 10, 12, 13, 14, 1, 5, 10, 14, 17, 18, 19, 20, 1, 5, 12, 16, 21, 23, 25, 26, 27, 1, 6, 14, 20, 27, 29, 32, 34, 35, 36, 1, 6, 16, 22, 32, 35, 40, 43, 45, 46, 47, 1, 7, 19, 28, 40, 45, 52, 57, 60, 62, 63, 64

Offset: 1

Alois P. Heinz, Aug 14 2012

Differs from A026820 first at T(7,4).

Differs from A213008 first at T(8,3).

			T(3,2) =  2 = |{3!/(2!*1!), 3!/(1!*1!*1!)}| = |{3, 6}|.
T(5,2) =  3 = |{30, 60, 120}|.
T(7,4) = 10 = |{35, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040}|.
T(8,3) = 10 = |{560, 1120, 1680, 2520, 3360, 5040, 6720, 10080, 20160, 40320}|.
T(9,2) =  5 = |{22680, 45360, 90720, 181440, 362880}|.
Triangle T(n,k) begins:
  1;
  1,  2;
  1,  2,  3;
  1,  3,  4,  5;
  1,  3,  5,  6,  7;
  1,  4,  7,  9, 10, 11;
  1,  4,  8, 10, 12, 13, 14;
  1,  5, 10, 14, 17, 18, 19, 20;
  1,  5, 12, 16, 21, 23, 25, 26, 27;
  1,  6, 14, 20, 27, 29, 32, 34, 35, 36;

Alois P. Heinz, Rows n = 1..75, flattened
Wikipedia, Multinomial coefficients

Main diagonal gives: A070289.
Columns k=1-3 give: A000012, A008619 (for n>1), A001399 (for n>2).
Cf. A026820, A213008, A215521.

b:= proc(n, i) option remember; `if`(n=0, {1}, `if`(i<1, {},
      {b(n, i-1)[], seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)}))
    end:
T:= (n, k)-> nops(b(n, k)):
seq(seq(T(n, k), k=1..n), n=1..14);

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, Join[b[n, i - 1], Table[ b[n - i*j, i - 1] *i!^j, {j, 1, n/i}] // Flatten]] // Union]; T[n_, k_] := Length[b[n, k]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 21 2015, after Alois P. Heinz *)

A382025 Triangle read by rows: T(n, k) is the number of partitions of n with at most k parts where 0 <= k <= n, and each part is one of three kinds.

Original entry on oeis.org

1, 0, 3, 0, 3, 9, 0, 3, 12, 22, 0, 3, 18, 36, 51, 0, 3, 21, 57, 87, 108, 0, 3, 27, 82, 148, 193, 221, 0, 3, 30, 111, 225, 330, 393, 429, 0, 3, 36, 144, 333, 528, 681, 765, 810, 0, 3, 39, 184, 460, 808, 1106, 1316, 1424, 1479, 0, 3, 45, 225, 630, 1182, 1740, 2163, 2439, 2574, 2640

Offset: 0

Peter Dolland, Mar 12 2025

The 1-kind case is Euler's table A026820.

The 2-kind case is A381895.

			Triangle starts:
 0 : [1]
 1 : [0, 3]
 2 : [0, 3,  9]
 3 : [0, 3, 12,  22]
 4 : [0, 3, 18,  36,  51]
 5 : [0, 3, 21,  57,  87,  108]
 6 : [0, 3, 27,  82, 148,  193,  221]
 7 : [0, 3, 30, 111, 225,  330,  393,  429]
 8 : [0, 3, 36, 144, 333,  528,  681,  765,  810]
 9 : [0, 3, 39, 184, 460,  808, 1106, 1316, 1424, 1479]
10 : [0, 3, 45, 225, 630, 1182, 1740, 2163, 2439, 2574, 2640]
...

Main diagonal gives A000716.

Cf. A026820, A381895.

from sympy import binomial
from sympy.utilities.iterables import partitions
from sympy.combinatorics.partitions import IntegerPartition
kinds = 3 - 1   # the number of part kinds - 1
def a382025_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        p = IntegerPartition( p).as_dict()
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( kinds + p[k], kinds)
        if s > 0 :
            t[s - 1] += fact
    for i in range( n - 1):
        t[i+1] += t[i]
    return [0] + t

A382045 Triangle read by rows: T(n,k) is the number of partitions of a 3-colored set of n objects into at most k parts with 0 <= k <= n.

Original entry on oeis.org

1, 0, 3, 0, 6, 12, 0, 10, 28, 38, 0, 15, 66, 102, 117, 0, 21, 126, 249, 309, 330, 0, 28, 236, 562, 788, 878, 906, 0, 36, 396, 1167, 1845, 2205, 2331, 2367, 0, 45, 651, 2292, 4128, 5289, 5814, 5982, 6027, 0, 55, 1001, 4272, 8703, 12106, 13881, 14602, 14818, 14873, 0, 66, 1512, 7608, 17634, 26616, 32088, 34608, 35556, 35826, 35892

Offset: 0

Peter Dolland, Mar 13 2025

The 1-color case is Euler's table A026820.

The 2-color case is A381891.

			Triangle starts:
 0 : [1]
 1 : [0,  3]
 2 : [0,  6,   12]
 3 : [0, 10,   28,   38]
 4 : [0, 15,   66,  102,   117]
 5 : [0, 21,  126,  249,   309,   330]
 6 : [0, 28,  236,  562,   788,   878,   906]
 7 : [0, 36,  396, 1167,  1845,  2205,  2331,  2367]
 8 : [0, 45,  651, 2292,  4128,  5289,  5814,  5982,  6027]
 9 : [0, 55, 1001, 4272,  8703, 12106, 13881, 14602, 14818, 14873]
10 : [0, 66, 1512, 7608, 17634, 26616, 32088, 34608, 35556, 35826, 35892]
...

Alois P. Heinz, Rows n = 0..150, flattened

Main diagonal gives A217093.

Cf. A026820, A381891, A000217.

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1))*binomial(i*(i+3)/2+j, j)*x^j, j=0..n/i))))
    end:
T:= proc(n, k) option remember;
     `if`(k<0, 0, T(n, k-1)+coeff(b(n$2), x, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 13 2025

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1]]*Binomial[i*(i + 3)/2 + j, j]*x^j, {j, 0, n/i}]]]];
T[n_, k_] := T[n, k] = If[k < 0, 0, T[n, k-1] + Coefficient[b[n, n], x, k]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 21 2025, after Alois P. Heinz *)

from sympy import binomial
from sympy.utilities.iterables import partitions
from sympy.combinatorics.partitions import IntegerPartition
colors = 3 - 1   # the number of colors - 1
def t_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        p = IntegerPartition( p).as_dict()
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( binomial( k + colors, colors) + p[k] - 1, p[k])
        if s > 0 :
            t[s - 1] += fact
    for i in range( n - 1):
        t[i+1] += t[i]
    return [0] + t

T(n,1) = binomial(n + 2, 2) = A000217(n + 1) for n >= 1.

cp's OEIS Frontend

A117488 The number 1 followed by 2*k+1 terms from column k of table A115994.

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A097306 Array of number of partitions of n with odd parts not exceeding 2*m-1 with m in {1, 2, ..., ceiling(n/2)}.

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A249770 Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order n with k invariant factors (2 <= n, 1 <= k).

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A249771 Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order A025487(n) with k invariant factors (2 <= n, 1 <= k).

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A008640 Number of partitions of n into at most 11 parts.

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A008641 Number of partitions of n into at most 12 parts.

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A134737 Number of partitions of the n-th partition number into positive parts not greater than n.

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A215520 Number T(n,k) of distinct values of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n with largest part <= k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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