A000712 -id:A000712 - cp's OEIS frontend

A382342 Triangle read by rows: T(n, k) is the number of partitions of n into k parts where 0 <= k <= n, and each part is one of two kinds.

1, 0, 2, 0, 2, 3, 0, 2, 4, 4, 0, 2, 7, 6, 5, 0, 2, 8, 12, 8, 6, 0, 2, 11, 18, 17, 10, 7, 0, 2, 12, 26, 28, 22, 12, 8, 0, 2, 15, 34, 46, 38, 27, 14, 9, 0, 2, 16, 46, 64, 66, 48, 32, 16, 10, 0, 2, 19, 56, 94, 100, 86, 58, 37, 18, 11, 0, 2, 20, 70, 124, 152, 136, 106, 68, 42, 20, 12

Offset: 0

Peter Dolland, Mar 27 2025

			Triangle starts:
 0 : [1]
 1 : [0, 2]
 2 : [0, 2,  3]
 3 : [0, 2,  4,  4]
 4 : [0, 2,  7,  6,  5]
 5 : [0, 2,  8, 12,  8,   6]
 6 : [0, 2, 11, 18, 17,  10,  7]
 7 : [0, 2, 12, 26, 28,  22, 12,  8]
 8 : [0, 2, 15, 34, 46,  38, 27, 14,  9]
 9 : [0, 2, 16, 46, 64,  66, 48, 32, 16, 10]
10 : [0, 2, 19, 56, 94, 100, 86, 58, 37, 18, 11]
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Row sums give A000712.

Cf. A008284 (1-kind case), A022597, A381895, A382345.

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(x^j*b(n-i*j, min(n-i*j, i-1))*(j+1), j=0..n/i))))
    end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=0..n), n=0..11);  # Alois P. Heinz, Mar 27 2025

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[x^j*b[n - i*j, Min[n - i*j, i - 1]]*(j + 1), {j, 0, n/i}]]]];
T[n_, k_] := Coefficient[b[n, n], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 11}] // Flatten (* Jean-François Alcover, Apr 19 2025, after Alois P. Heinz *)

from sympy.utilities.iterables import partitions
def t_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= 1 + p[k]
        if s > 0 :
            t[s - 1] += fact
    return [0] + t

T(n,n) = n + 1.
T(n,1) = 2 for n >= 1.
T(n,k) = A381895(n,k) - A381895(n,k-1) for 1 <= k <= n.
Sum_{k=0..n} (-1)^k * T(n,k) = A022597(n). - Alois P. Heinz, Mar 27 2025

A382345 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where n unlabeled objects are distributed into k containers of two kinds. Containers may be left empty.

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 4, 2, 0, 5, 6, 7, 2, 0, 6, 8, 12, 8, 2, 0, 7, 10, 17, 18, 11, 2, 0, 8, 12, 22, 28, 26, 12, 2, 0, 9, 14, 27, 38, 46, 34, 15, 2, 0, 10, 16, 32, 48, 66, 64, 46, 16, 2, 0, 11, 18, 37, 58, 86, 100, 94, 56, 19, 2, 0, 12, 20, 42, 68, 106, 136, 152, 124, 70, 20, 2, 0

Offset: 0

Peter Dolland, Mar 29 2025

			Array starts:
 0 : [1, 2,  3,   4,   5,   6,   7,    8,    9,   10,   11]
 1 : [0, 2,  4,   6,   8,  10,  12,   14,   16,   18,   20]
 2 : [0, 2,  7,  12,  17,  22,  27,   32,   37,   42,   47]
 3 : [0, 2,  8,  18,  28,  38,  48,   58,   68,   78,   88]
 4 : [0, 2, 11,  26,  46,  66,  86,  106,  126,  146,  166]
 5 : [0, 2, 12,  34,  64, 100, 136,  172,  208,  244,  280]
 6 : [0, 2, 15,  46,  94, 152, 217,  282,  347,  412,  477]
 7 : [0, 2, 16,  56, 124, 214, 316,  426,  536,  646,  756]
 8 : [0, 2, 19,  70, 167, 302, 464,  640,  825, 1010, 1195]
 9 : [0, 2, 20,  84, 212, 406, 648,  922, 1212, 1512, 1812]
10 : [0, 2, 23, 100, 271, 542, 899, 1314, 1766, 2236, 2717]
...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Antidiagonal sums give A000712.
Alternating antidiagonal sums give A073252.
Without empty containers: A381895.
Cf. A382342.

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(x^j*b(n-i*j, min(n-i*j, i-1))*(j+1), j=0..n/i))))
    end:
A:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(A(n, d-n), n=0..d), d=0..11);  # Alois P. Heinz, Mar 29 2025

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
   Sum[x^j*b[n - i*j, Min[n - i*j, i - 1]]*(j + 1), {j, 0, n/i}]]]];
A[n_, k_] := Coefficient[b[n + k, n + k], x, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Apr 07 2025, after Alois P. Heinz *)

from sympy.utilities.iterables import partitions
def a_row(n, length=11) -> list[int]:
    if n == 0 : return list(range(1, length + 1))
    t = [0] * length
    for p in partitions(n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= p[k] + 1
        if s > 0 :
            t[s] += fact
    for i in range(1, length - 1):
        t[i+1] += t[i] * 2 - t[i-1]
    return t
for n in range(11): print(a_row(n))

A(0,k) = k + 1.
A(1,k) = 2*k.
A(2,k+1) = 2 + 5 * k.
A(n,1) = 2.
A(n,k) = Sum_{i=0..k} (k + 1 - i) * A382342(n,i) for k <= n.
A(n,n+k) = A(n,n) + k * A000712(n).
A(n,k) = A382342(n,k) + 2 * A(n,k-1) - A(n,k-2) for 2 <= k <= n.
A(n,k) = A382342(n+k,k). - Alois P. Heinz, Mar 31 2025

A061160 Numerators in expansion of Euler transform of b(n) = 1/3.

Original entry on oeis.org

1, 1, 5, 50, 215, 646, 8711, 25475, 105925, 3091270, 11691247, 36809705, 445872155, 1364113925, 5085042010, 50975292560, 183383680088, 588817265695, 19512559194875, 62369303509475, 224877933068647, 2214198452392027, 7686538660149565, 25124342108522750

Offset: 0

Vladeta Jovovic, Apr 17 2001

Denominators of c(n) are 3^d(n), where d(n)=power of 3 in (3*n)!, cf. A004128.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Geoffrey B. Campbell, Some n-space q-binomial theorem extensions and similar identities, arXiv:1906.07526 [math.NT], 2019.
Geoffrey B. Campbell, Continued Fractions for partition generating functions, arXiv:2301.12945 [math.CO], 2023.
Geoffrey B. Campbell, Vector Partition Identities for 2D, 3D and nD Lattices, arXiv:2302.01091 [math.CO], 2023.
Geoffrey B. Campbell and A. Zujev, Some almost partition theoretic identities, Preprint, 2016.
N. J. A. Sloane, Transforms

Cf. A000712, A061159, A061161.

b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d/3, d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 28 2017

c[n_] := c[n] = If[n == 0, 1,
     (1/(3n)) Sum[c[n-k] DivisorSigma[1, k], {k, 1, n}]];
a[n_] := Numerator[c[n]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 24 2022 *)

Numerators of c(n), where c(n) = (1/(3*n))*Sum_{k=1..n} c(n-k)*sigma(k), n>0, c(0)=1.

A146023 Triangle read by rows, square of A027293.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 10, 5, 2, 1, 20, 10, 5, 2, 1, 36, 20, 10, 5, 2, 1, 65, 36, 20, 10, 5, 2, 1, 110, 65, 36, 20, 10, 5, 2, 1, 185, 110, 65, 36, 20, 10, 5, 2, 1, 300, 185, 110, 65, 36, 20, 10, 5, 2, 1

Offset: 0

Gary W. Adamson, Oct 26 2008

Triangle, A000712 (1, 2, 5, 10, 20, 36,...) in every column.
Row sums = A000713: (1, 3, 8, 18, 38, 74, 139,...)
Note that we are working here in the world of lower triangular matrices. - N. J. A. Sloane, Jun 15 2020

			First few rows of the triangle =
    1;
    2,  1;
    5,  2,  1;
   10,  5,  2,  1;
   20, 10,  5,  2,  1;
   36, 20, 10,  5,  2,  1;
   65, 36, 20, 10,  5,  2,  1;
  110, 65, 36, 20, 10,  5,  2,  1;
...

Robert Price, Table of n, a(n) for n = 0..819 (first 40 rows)

Cf. A000712, A000713, A027293.

lim = 10;
A000712 = Table [Length@IntegerPartitions[n, All, Range@n~Join~Range@n], {n, 0, lim - 1}]
Table[Reverse[Take[A000712, n]], {n, lim}] // Flatten (* Robert Price, Jun 15 2020 *)

Extraneous data deleted by Robert Price, Jun 15 2020

A234937 Triangle read by rows of coefficients of polynomials generated by the Han/Nekrasov-Okounkov formula.

Original entry on oeis.org

1, 1, -1, 4, -5, 1, 18, -29, 12, -1, 120, -218, 119, -22, 1, 840, -1814, 1285, -345, 35, -1, 7920, -18144, 14674, -5205, 805, -51, 1, 75600, -196356, 185080, -79219, 16450, -1624, 70, -1, 887040, -2427312, 2515036, -1258628, 324569, -43568, 2954, -92, 1

Offset: 0

William J. Keith, Jan 01 2014

Coefficients of the polynomials p_n(b) defined by Product_{k>0} (1-q^k)^(b-1) = Sum n! p_n(b) q^n.
Each row is length 1+n, starting from n=0, and consists of the coefficients of one of the p_n(b).
A210590 is an unsigned version using the form preferred by Nekrasov and Okounkov. This is the form for which Guo-Niu Han's reference below gives the hooklength formula:
p_n(b) = Sum_{lambda partitioning n} Product_{h_{ij} in lambda} (1-b/(h_{ij}^2)).
Coefficients reduced mod 5 are those of 2 times Pascal's triangle and an alternating sign. Other primes have slightly more complex reduction behavior. See second link.
Lehmer's conjecture on the tau function states that the evaluation at b=25 (A000594) is never 0.
The general diagonal and column are probably of combinatorial interest.

			The coefficient of q^3 in the indeterminate power is (1/6) (18-29b+12b^2-b^3).

Seiichi Manyama, Rows n = 0..100, flattened
G.-N. Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.
W. J. Keith, Polynomial analogues of Ramanujan congruences for Han's hooklength formula, arXiv:1109.1236 [math.CO], 2011-2012; Acta Arith. 160 (2013), 303-315.

Row entries sum to 0.
A210590 is the unsigned version.
Starting from row 0: final entry of row n, (-1)^n (A033999).
From row 1: next-to-last entry of row n, (-1)^(n-1) * n(3n-1)/2 (signed version of A000326).
First entry of row n, n! * p(n) (A053529).
Second entry of row n, -1 * n! * (sum of reciprocals of all parts in partitions of n) (negatives of A057623).
(Sum of absolute values of row entries)/n!: A000712.
Evaluations at various powers of b, divided by n!, enumerate multipartitions or powers of the eta function. Some special cases that appear in the OEIS:
b=0: A000041, the partition numbers,
b=2: A010815, from Euler's Pentagonal Number Theorem,
b=-1: A000712, partitions into 2 colors,
b=-11: A005758, reciprocal of the square root of the tau function,
b=-23: A006922, reciprocal of the tau function,
b=13: A000735, square root of the tau function,
b=25: A000594, Ramanujan's tau function.

nn=10;
Clear[b]; PolyTable = Table[0, {n, 1, nn}];
PolyTable[[1]]=1-b;
For[n = 2, n <= nn, n++,
PolyTable[[n]] = Simplify[(((n - 1)!)*(b - 1))*(Sum[
       PolyTable[[n - m]]*(-1*DivisorSigma[1, m]/((n - m)!)), {m, 1,
        n - 1}] + (-1*DivisorSigma[1, n]))]];
LongTable = Table[Table[
   Which[k == 0, PartitionsP[n]*n!, k > 0,
    Coefficient[Expand[PolyTable[[n]]], b^k]], {k, 0, n}], {n, 1, nn}];
Flatten[PrependTo[LongTable,1]]

E.g.f.: Product_{k>0} (1-q^k)^(b-1).

Recurrence: With p_0(b) = 1, p_n(b) = (n-1)!*(b-1)*Sum_{m=1..n} -sigma(m)*p_{n-m}(b) / (n-m)!, sigma being the divisor function.

A241153 Number of partitions having the maximal degree in the partition graph G(n) defined at A241150.

Original entry on oeis.org

2, 1, 1, 2, 1, 1, 2, 5, 1, 1, 2, 5, 10, 1, 1, 2, 5, 10, 20, 1, 1, 2, 5, 10, 20, 36, 1, 1, 2, 5, 10, 20, 36, 65, 1, 1, 2, 5, 10, 20, 36, 65, 110, 1, 1, 2, 5, 10, 20, 36, 65, 110, 185, 1, 1, 2, 5, 10, 20, 36, 65, 110, 185, 300

Offset: 2

Clark Kimberling and Peter J. C. Moses, Apr 17 2014

a(n) = last number in row n of G(n), for n >= 2.  The numbers in this sequence can be formatted as a triangle:
    2
1 2
1 2 5
1 2 5 10
1 2 5 10 20
1 2 5 10 20 36 ...
    Deleting column 1 leaves
2
2 5
2 5 10
2 5 10 20
2 5 10 20 36... ,
    in which row n is identical to the first n+1 terms of A000712.

			a(9) counts these 5 partitions:  5211, 4311, 42111, 321111, 32211, which all have degree 5, which is maximal for the graph G(9), as seen by putting k = 9 in the Mathematica program.  (See the Example section of A241150.)

Cf. A241150, A241151, A241152.

z = 25; spawn[part_] := Map[Reverse[Sort[Flatten[ReplacePart[part, {# - 1, 1}, Position[part, #, 1, 1][[1]][[1]]]]]] &, DeleteCases[DeleteDuplicates[part], 1]];
     unspawn[part_] := If[Length[Cases[part, 1]] > 0, Map[ReplacePart[Most[part], Position[Most[part], #, 1, 1][[1]][[1]] -> # + 1] &, DeleteDuplicates[Most[part]]], {}]; m = Map[Last[Transpose[Tally[Map[#[[2]] &, Tally[Flatten[{Map[unspawn, #], Map[spawn, #]}, 2] &[IntegerPartitions[#]]]]]]] &, 1 + Range[z]];
     Column[m] (* A241150 as an array *)
     Flatten[m] (* A241150 as a sequence *)
     Table[Length[m[[n]]], {n, 1, z}] (* A241151 *)
     Table[Max[m[[n]]], {n, 1, z}] (* A241152 *)
     Table[Last[m[[n]]], {n, 1, z}] (* A241153 *)
     (* Next, show the graph G(k) *)
     k = 8; graph = Flatten[Table[part = IntegerPartitions[k][[n]]; Map[FromDigits[part] -> FromDigits[#] &, spawn[part]], {n, 1, PartitionsP[k]}]]; Graph[graph, VertexLabels -> "Name", ImageSize -> 500, ImagePadding -> 20] (* Peter J. C. Moses, Apr 15 2014 *)

A298434 Expansion of Product_{k>=1} 1/(1 - x^(k^3))^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 14, 17, 20, 23, 26, 29, 32, 38, 44, 50, 56, 62, 68, 74, 80, 90, 100, 110, 122, 134, 146, 158, 170, 187, 204, 221, 242, 263, 284, 305, 326, 353, 380, 407, 440, 473, 506, 539, 572, 612, 652, 692, 740, 788, 836, 887, 938, 997, 1056, 1115, 1184, 1253

Offset: 0

Ilya Gutkovskiy, Jan 19 2018

nonn

Number of partitions of n into cubes of 2 kinds.

Self-convolution of A003108.

			a(8) = 11 because we have [8a], [8b], [1a, 1a, 1a, 1a, 1a, 1a, 1a, 1a], [1a, 1a, 1a, 1a, 1a, 1a, 1a, 1b], [1a, 1a, 1a, 1a, 1a, 1a, 1b, 1b], [1a, 1a, 1a, 1a, 1a, 1b, 1b, 1b], [1a, 1a, 1a, 1a, 1b, 1b, 1b, 1b], [1a, 1a, 1a, 1b, 1b, 1b, 1b, 1b], [1a, 1a, 1b, 1b, 1b, 1b, 1b, 1b], [1a, 1b, 1b, 1b, 1b, 1b, 1b, 1b] and [1b, 1b, 1b, 1b, 1b, 1b, 1b, 1b].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000578, A000712, A003108, A279225.

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(k^3))^2, {k, 1, Floor[nmax^(1/3) + 1]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 08 2018 *)

G.f.: Product_{k>=1} 1/(1 - x^(k^3))^2.

a(n) ~ exp(2^(11/4) * (Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3) * Zeta(4/3))^(9/8) / (2^(27/8) * 3^(7/4) * Pi^(7/2) * n^(13/8)). - Vaclav Kotesovec, Apr 08 2018

A298435 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)/2))^2.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 20, 28, 36, 52, 70, 88, 120, 156, 192, 250, 318, 386, 488, 606, 727, 900, 1101, 1308, 1590, 1916, 2257, 2706, 3225, 3768, 4465, 5270, 6117, 7178, 8399, 9686, 11274, 13094, 15020, 17352, 20017, 22846, 26230, 30080, 34175, 39010, 44500, 50346, 57184, 64914, 73156

Offset: 0

Ilya Gutkovskiy, Jan 19 2018

nonn

Number of partitions of n into triangular numbers of 2 kinds.

Self-convolution of A007294.

			a(3) = 6 because we have [3a], [3b], [1a, 1a, 1a], [1a, 1a, 1b], [1a, 1b, 1b] and [1b, 1b, 1b].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000217, A000712, A007294, A279225.

nmax = 50; CoefficientList[Series[Product[1/(1 - x^(k (k + 1)/2))^2, {k, 1, nmax}], {x, 0, nmax}], x]

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

a(n) ~ exp(3*(Pi/2)^(1/3) * Zeta(3/2)^(2/3) * n^(1/3)) * Zeta(3/2)^(5/3) / (2^(29/6) * sqrt(3) * Pi^(5/3) * n^(13/6)). - Vaclav Kotesovec, Apr 08 2018

A301934 Number of positive subset-sum trees of weight n.

Original entry on oeis.org

1, 3, 14, 85, 586, 4331, 33545, 268521, 2204249

Offset: 1

Gus Wiseman, Mar 28 2018

A positive subset-sum tree with root x is either the symbol x itself, or is obtained by first choosing a positive subset-sum x <= (y_1,...,y_k) with k > 1 and then choosing a positive subset-sum tree with root y_i for each i = 1...k. The weight is the sum of the leaves. We write positive subset-sum trees in the form rootsum(branch,...,branch). For example, 4(1(1,3),2,2(1,1)) is a positive subset-sum tree with composite 4(1,1,1,2,3) and weight 8.

			The a(3) = 14 positive subset-sum trees:
3           3(1,2)       3(1,1,1)     3(1,2(1,1))
2(1,2)      2(1,1,1)     2(1,1(1,1))  2(1(1,1),1)  2(1,2(1,1))
1(1,2)      1(1,1,1)     1(1,1(1,1))  1(1(1,1),1)  1(1,2(1,1))

Cf. A000108, A000712, A108917, A122768, A262671, A262673, A275972, A276024, A284640, A299701, A301854, A301855, A301856, A301935.

A327215 Self-convolution of A008485.

Original entry on oeis.org

1, 2, 11, 54, 279, 1442, 7530, 39474, 207693, 1095522, 5790116, 30650038, 162451560, 861920492, 4577055823, 24323292984, 129338944225, 688128700422, 3662798123481, 19504467792378, 103899170100154, 553642311668244, 2951010332435759, 15733439067954134

Offset: 0

Vaclav Kotesovec, Aug 26 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000712, A008485, A304444, A309986, A327214.

A008485[n_]:=SeriesCoefficient[Product[1/(1-x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A008485[k]*A008485[n-k], {k, 0, n}], {n, 0, 25}]

a(n) ~ c^2 * Pi * d^n, where d = A270915 = 5.352701333486642687772415814165... and c = A327279 = 0.26801521271073331568695383828... (see A008485).

cp's OEIS Frontend

A382342 Triangle read by rows: T(n, k) is the number of partitions of n into k parts where 0 <= k <= n, and each part is one of two kinds.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A382345 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where n unlabeled objects are distributed into k containers of two kinds. Containers may be left empty.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A061160 Numerators in expansion of Euler transform of b(n) = 1/3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A146023 Triangle read by rows, square of A027293.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A234937 Triangle read by rows of coefficients of polynomials generated by the Han/Nekrasov-Okounkov formula.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A241153 Number of partitions having the maximal degree in the partition graph G(n) defined at A241150.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A298434 Expansion of Product_{k>=1} 1/(1 - x^(k^3))^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A298435 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)/2))^2.

Views