A284593 -id:A284593 - cp's OEIS frontend

A032302 G.f.: Product_{k>=1} (1 + 2*x^k).

1, 2, 2, 6, 6, 10, 18, 22, 30, 42, 66, 78, 110, 138, 186, 254, 318, 402, 522, 654, 822, 1074, 1306, 1638, 2022, 2514, 3058, 3798, 4662, 5658, 6882, 8358, 10062, 12186, 14610, 17534, 21150, 25146, 29994, 35694, 42446, 50178, 59514, 70110, 82758, 97602, 114570, 134262

Offset: 0

Christian G. Bower, Apr 01 1998

nonn

"EFK" (unordered, size, unlabeled) transform of 2,2,2,2,...
Number of partitions into distinct parts of 2 sorts, see example. - Joerg Arndt, May 22 2013
In general, for a fixed integer m > 0, if g.f. = Product_{k>=1} (1 + m*x^k) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m+1)*Pi)*n^(3/4)), where c = Pi^2/6 + log(m)^2/2 + polylog(2, -1/m) = -polylog(2, -m). - Vaclav Kotesovec, Jan 04 2016
Antidiagonal sums of A284593. - Peter Bala, Mar 30 2017

			From _Joerg Arndt_, May 22 2013: (Start)
There are a(7) = 22 partitions of 7 into distinct parts of 2 sorts (format P:S for part:sort):
01:  [ 1:0  2:0  4:0  ]
02:  [ 1:0  2:0  4:1  ]
03:  [ 1:0  2:1  4:0  ]
04:  [ 1:0  2:1  4:1  ]
05:  [ 1:0  6:0  ]
06:  [ 1:0  6:1  ]
07:  [ 1:1  2:0  4:0  ]
08:  [ 1:1  2:0  4:1  ]
09:  [ 1:1  2:1  4:0  ]
10:  [ 1:1  2:1  4:1  ]
11:  [ 1:1  6:0  ]
12:  [ 1:1  6:1  ]
13:  [ 2:0  5:0  ]
14:  [ 2:0  5:1  ]
15:  [ 2:1  5:0  ]
16:  [ 2:1  5:1  ]
17:  [ 3:0  4:0  ]
18:  [ 3:0  4:1  ]
19:  [ 3:1  4:0  ]
20:  [ 3:1  4:1  ]
21:  [ 7:0  ]
22:  [ 7:1  ]
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Moussa Ahmia and Mohammed L. Nadji, Overpartitions into distinct parts, Comb. Number Theory 14 (2025), no. 1, 65-74, preprint (2021).
C. G. Bower, Transforms (2)
Vaclav Kotesovec, Asymptotic formula for A032302
Eric Weisstein's World of Mathematics, Dilogarithm
Eric Weisstein's MathWorld, Polylogarithm
Wikipedia, Polylogarithm

Cf. A000009, A032308, A261562, A261568, A261569, A266576, A284593.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 2*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015
# Alternatively:
simplify(expand(QDifferenceEquations:-QPochhammer(-2,x,99)/3,x)):
seq(coeff(%,x,n), n=0..47); # Peter Luschny, Nov 17 2016

nn=47; CoefficientList[Series[Product[1+2x^i,{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 07 2013 *)
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*2^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-2, x]/3 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

N=66; x='x+O('x^N); Vec(prod(n=1,N, 1+2*x^n)) \\ Joerg Arndt, May 22 2013

a(n) = A072706(n)*2 for n>=1.
G.f.: Sum_{n>=0} (2^n*q^(n*(n+1)/2) / Product_{k=1..n} (1-q^k ) ). - Joerg Arndt, Jan 20 2014
a(n) = (1/3) [x^n] QPochhammer(-2,x). - Vladimir Reshetnikov, Nov 20 2015
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Pi^2/6 + log(2)^2/2 + polylog(2, -1/2) = 1.43674636688368094636290202389358335424... . Equivalently, c = A266576 = Pi^2/12 + log(2)^2 + polylog(2, 1/4)/2. - Vaclav Kotesovec, Jan 04 2016

A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 1, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 1, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 12, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12

Offset: 0

Gus Wiseman, Sep 16 2023

First differs from A284593 at T(6,3) = 1, A284593(6,3) = 2.
Rows are palindromic.
Are there only two zeros in the whole triangle?

			Triangle begins:
  1
  1  1
  1  0  1
  2  1  1  2
  2  1  0  1  2
  3  1  1  1  1  3
  4  2  2  1  2  2  4
  5  2  2  2  2  2  2  5
  6  3  2  3  1  3  2  3  6
  8  3  3  4  3  3  4  3  3  8
Row n = 6 counts the following strict partitions:
  (6)      (5,1)    (4,2)    (3,2,1)  (4,2)    (5,1)    (6)
  (5,1)    (3,2,1)  (3,2,1)           (3,2,1)  (3,2,1)  (5,1)
  (4,2)                                                 (4,2)
  (3,2,1)                                               (3,2,1)
Row n = 10 counts the following strict partitions:
  A     91    82    73    64    532   64    73    82    91    A
  64    541   532   532   541   541   541   532   532   541   64
  73    631   721   631   631   4321  631   631   721   631   73
  82    721   4321  721   4321        4321  721   4321  721   82
  91    4321        4321                    4321        4321  91
  532                                                         532
  541                                                         541
  631                                                         631
  721                                                         721
  4321                                                        4321

Robert Price, Table of n, a(n) for n = 0..1325

Columns k = 0 and k = n are A000009.
The non-strict complement is A046663, central column A006827.
Central column n = 2k is A237258.
For subsets instead of partitions we have A365381.
The non-strict case is A365543.
The complement is A365663.
A000124 counts distinct possible sums of subsets of {1..n}.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Table[Length[Select[Select[IntegerPartitions[n], UnsameQ@@#&], MemberQ[Total/@Subsets[#],k]&]], {n,0,10},{k,0,n}]

A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 33, 46, 68, 95, 140, 187, 266, 372, 507, 683, 948, 1256, 1692, 2263, 3003, 3955, 5248, 6824, 8921, 11669, 15058, 19413, 25128, 32149, 41129, 52578, 66740, 84696, 107389, 135310, 170277, 214386, 268151, 335261, 418896, 521204

Offset: 0

Wouter Meeussen, Jul 09 2005

nonn

Counted as orderless pairs since intersection is commutative.

			Of the partitions of 12 into different parts, the partition (5+4+2+1) has an empty intersection with only (12) and (9+3).
From _Gus Wiseman_, Oct 07 2023: (Start)
The a(6) = 4 pairs are:
  ((6),(5,1))
  ((6),(4,2))
  ((6),(3,2,1))
  ((5,1),(4,2))
(End)

Vaclav Kotesovec Table of n, a(n) for n = 0..1050 (terms 0..700 from Alois P. Heinz)

Column k=2 of A258280.
Cf. A079122, A086737.
Main diagonal of A284593 times (1/2).
This is the strict case of A260669.
The ordered version is A365662 = strict case of A054440.
This is the disjoint case of A366132, with twins A366317.
A000041 counts integer partitions, strict A000009.
A002219 counts biquanimous partitions, strict A237258, ordered A064914.
Cf. A000124, A000712, A001255, A006827, A032302, A122768, A355389.

using DiscreteMath`Combinatorica`and ListPartitionsQ[n_Integer]:= Flatten[ Reverse /@ Table[(Range[m-1, 0, -1]+#1&)/@ TransposePartition/@ Complement[Partitions[ n-m* (m-1)/2, m], Partitions[n-m*(m-1)/2, m-1]], {m, -1+Floor[1/2*(1+Sqrt[1+8*n])]}], 1]; Table[Plus@@Flatten[Outer[If[Intersection[Flatten[ #1], Flatten[ #2]]==={}, 1, 0]&, ListPartitionsQ[k], ListPartitionsQ[k], 1]], {k, 48}]/2
nmax = 50; p = 1; Do[p = Expand[p*(1 + x^j + y^j)]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {j, 1, nmax}]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Table[Coefficient[p, x^n*y^n]/2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 07 2017 *)
Table[Length[Select[Subsets[Select[IntegerPartitions[n], UnsameQ@@#&],{2}],Intersection@@#=={}&]],{n,15}] (* Gus Wiseman, Oct 07 2023 *)

a(n) = {my(A=1 + O(x*x^n) + O(y*y^n)); polcoef(polcoef(prod(k=1, n, A + x^k + y^k), n), n)/2} \\ Andrew Howroyd, Oct 10 2023

a(n) = ceiling(1/2 * [(x*y)^n] Product_{j>0} (1+x^j+y^j)). - Alois P. Heinz, Mar 31 2017

a(n) = ceiling(A365662(n)/2). - Gus Wiseman, Oct 07 2023

Name edited by Gus Wiseman, Oct 10 2023

a(0)=1 prepended by Alois P. Heinz, Feb 09 2024

A322210 G.f.: P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)), where P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k, as a square table of coefficients T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 3, 5, 7, 10, 7, 5, 7, 12, 18, 18, 12, 7, 11, 19, 34, 38, 34, 19, 11, 15, 30, 56, 74, 74, 56, 30, 15, 22, 45, 94, 133, 158, 133, 94, 45, 22, 30, 67, 146, 233, 297, 297, 233, 146, 67, 30, 42, 97, 228, 385, 550, 602, 550, 385, 228, 97, 42, 56, 139, 340, 623, 951, 1166, 1166, 951, 623, 340, 139, 56

Offset: 0

Paul D. Hanna, Nov 30 2018

Conjecture 1: the triangular table T(n,k) is the number of ways to form the subsum k from the partitions of n, where n and k are integers such that 0 <= k <= n. For example, t(4,2)=10; the five partitions of 4 are (4), (3,1), (2,2), (2,1,1), (1,1,1,1) with subsum 2 occurring {0,0,2,2,6) times for a total of 10. - George Beck, Jan 03 2020
From Wouter Meeussen, Mar 09 2023: (Start)
Conjecture 2: the square table T(n,k) is the coefficient of s_lambda in the sum over all partitions lambda |-n and nu |-k of (s_rho/mu) where s_lambda*s_mu = Sum(rho|-n+k; C(rho, lambda, mu) s_rho). Simply stated as: multiply lambda with mu, and, for each term in the result, take the skew Schur function with mu and count how often you get the original lambda back. Sum up over all lambda and mu of the size n and k.
Conjecture 3: the triangular table T(n,k) is analogous to conjecture 2, but counting s_lambda in s_(lambda/mu) * s_mu with lambda |- n and mu |- k and 0<=k<=n. (End)

			G.f.: P(x,y) = 1 + (x + y) + (2*x^2 + 2*x*y + 2*y^2) + (3*x^3 + 4*x^2*y + 4*x*y^2 + 3*y^3) + (5*x^4 + 7*x^3*y + 10*x^2*y^2 + 7*x*y^3 + 5*y^4) + (7*x^5 + 12*x^4*y + 18*x^3*y^2 + 18*x^2*y^3 + 12*x*y^4 + 7*y^5) + (11*x^6 +19*x^5*y + 34*x^4*y^2 + 38*x^3*y^3 + 34*x^2*y^4 + 19*x*y^5 + 11*y^6) + (15*x^7 + 30*x^6*y + 56*x^5*y^2 + 74*x^4*y^3 + 74*x^3*y^4 + 56*x^2*y^5 + 30*x*y^6 + 15*y^7) + (22*x^8 + 45*x^7*y + 94*x^6*y^2 + 133*x^5*y^3 + 158*x^4*y^4 + 133*x^3*y^5 + 94*x^2*y^6 + 45*x*y^7 + 22*y^8) + ...
such that
P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)),
where
P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k.
SQUARE TABLE.
The square table of coefficients T(n,k) of x^n*y^k in P(x,y) begins
   1,   1,   2,    3,    5,     7,    11,     15,     22,     30, ...
   1,   2,   4,    7,   12,    19,    30,     45,     67,     97, ...
   2,   4,  10,   18,   34,    56,    94,    146,    228,    340, ...
   3,   7,  18,   38,   74,   133,   233,    385,    623,    977, ...
   5,  12,  34,   74,  158,   297,   550,    951,   1614,   2627, ...
   7,  19,  56,  133,  297,   602,  1166,   2133,   3775,   6437, ...
  11,  30,  94,  233,  550,  1166,  2382,   4551,   8424,  14953, ...
  15,  45, 146,  385,  951,  2133,  4551,   9142,  17639,  32680, ...
  22,  67, 228,  623, 1614,  3775,  8424,  17639,  35492,  68356, ...
  30,  97, 340,  977, 2627,  6437, 14953,  32680,  68356, 136936, ...
  42, 139, 506, 1501, 4202, 10692, 25835,  58659, 127443, 264747, ...
  56, 195, 730, 2255, 6531, 17290, 43313, 102149, 229998, 495195, ...
  ...
TRIANGLE.
Alternatively, this sequence may be written as a triangle, starting as
   1;
   1,   1;
   2,   2,   2;
   3,   4,   4,   3;
   5,   7,  10,   7,    5;
   7,  12,  18,  18,   12,    7;
  11,  19,  34,  38,   34,   19,   11;
  15,  30,  56,  74,   74,   56,   30,   15;
  22,  45,  94, 133,  158,  133,   94,   45,   22;
  30,  67, 146, 233,  297,  297,  233,  146,   67,  30;
  42,  97, 228, 385,  550,  602,  550,  385,  228,  97,  42;
  56, 139, 340, 623,  951, 1166, 1166,  951,  623, 340, 139,  56;
  77, 195, 506, 977, 1614, 2133, 2382, 2133, 1614, 977, 506, 195, 77;
  ...

Alois P. Heinz, Antidiagonals n = 0..200 (first 61 antidiagonals from Paul D. Hanna)

Cf. A322200 (log).
Cf. A000041 (row 0 = partitions), A000070 (row 1), A093695(k+2) (row 2).
Main diagonal gives A322211.
Antidiagonal sums give A070933.
Cf. A284593.
Cf. A361286.

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1,
      (x+1)^n, b(n, i-1) +(x^i+1)*b(n-i, min(n-i, i))))
    end:
T:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 23 2019

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (x + 1)^n, b[n, i - 1] + (x^i + 1) b[n - i, Min[n - i, i]]]];
T[n_, k_] := Coefficient[b[n + k, n + k], x, k];
Table[Table[T[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

{P = 1/prod(n=1,61, (1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{T(n,k) = polcoeff( polcoeff( P,n,x),k,y)}
for(n=0,16, for(k=0,16, print1( T(n,k),", ") );print(""))

FORMULAS FOR TERMS.
T(n,k) = T(k,n) for n >= 0, k >= 0.
T(n,0) = A000041(n) for n >= 0, where A000041 is the partition numbers.
T(n,1) = A000070(n) for n >= 0, where A000070 is the sum of partitions.
ROW GENERATING FUNCTIONS.
Row 0: 1/( Product_{n>=1} (1 - x^n) ).
Row 1: 1/( (1-x) * Product_{n>=1} (1 - x^n) ).
Row 2: 2/( (1-x) * (1-x^2) * Product_{n>=1} (1 - x^n) ).

A284592 Square array read by antidiagonals: T(n,k) is the number of pairs of partitions of n and k respectively, such that the pair of partitions have no part in common.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 3, 1, 1, 3, 5, 1, 2, 1, 5, 7, 2, 3, 3, 2, 7, 11, 2, 5, 4, 5, 2, 11, 15, 4, 6, 7, 7, 6, 4, 15, 22, 4, 10, 8, 12, 8, 10, 4, 22, 30, 7, 12, 14, 14, 14, 14, 12, 7, 30, 42, 8, 18, 16, 24, 16, 24, 16, 18, 8, 42, 56, 12, 23, 25, 28, 28, 28, 28, 25, 23, 12, 56

Offset: 0

Peter Bala, Mar 30 2017

Compare with A284593.

			Square array begins
  n\k|  0  1  2  3  4  5   6   7   8   9  10
- - - - - - - - - - - - - - - - - - - - - - -
  0  |  1  1  2  3  5  7  11  15  22  30  42: A000041
  1  |  1  0  1  1  2  2   4   4   7   8  12: A002865
  2  |  2  1  2  3  5  6  10  12  18  23  32
  3  |  3  1  3  4  7  8  14  16  25  31  44
  4  |  5  2  5  7 12 14  24  28  43  54  76
  5  |  7  2  6  8 14 16  28  31  49  60  85
  6  | 11  4 10 14 24 28  48  55  85 106 149
  7  | 15  4 12 16 28 31  55  60  95 115 163
  8  | 22  7 18 25 43 49  85  95 148 182 256
  9  | 30  8 23 31 54 60 106 115 182 220 311
  10 | 42 12 32 44 76 85 149 163 256 311 438
  ...
T(4,3) = 7: the 7 pairs of partitions of 4 and 3 with no parts in common are (4, 3), (4, 2 + 1), (4, 1 + 1 + 1), (2 + 2, 3), (2 + 2, 1 + 1 + 1), (2 + 1 + 1 , 3) and (1 + 1 + 1 + 1, 3).

Alois P. Heinz, Antidiagonals n = 0..200, flattened
H. S. Wilf, Lectures on Integer Partitions

Cf. A000041 (row 0), A002865 (row 1), A015128 (antidiagonal sums), A284593.

Main diagonal gives A054440 or 2*A260669 (for n>0).

#A284592 as a square array
ser := taylor(taylor(mul(1 + x^j/(1 - x^j) + y^j/(1 - y^j), j = 1..10), x, 11), y, 11):
convert(ser, polynom):
s := convert(%, polynom):
with(PolynomialTools):
for n from 0 to 10 do CoefficientList(coeff(s, y, n), x) end do;
# second Maple program:
b:= proc(n, k, i) option remember; `if`(n=0 and
       (k=0 or i=1), 1, `if`(i<1, 0, b(n, k, i-1)+
       add(b(sort([n-i*j, k])[], i-1), j=1..n/i)+
       add(b(sort([n, k-i*j])[], i-1), j=1..k/i)))
    end:
A:= (n, k)-> (l-> b(l[1], l[2]$2))(sort([n, k])):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Apr 02 2017

Table[Total@ Boole@ Map[! IntersectingQ @@ Map[Union, #] &, Tuples@ {IntegerPartitions@ #, IntegerPartitions@ k}] &[n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 02 2017 *)
b[n_, k_, i_] := b[n, k, i] = If[n == 0 &&
     (k == 0 || i == 1), 1, If[i < 1, 0, b[n, k, i - 1] +
     Sum[b[Sequence @@ Sort[{n - i*j, k}], i - 1], {j, 1, n/i}] +
     Sum[b[Sequence @@ Sort[{n, k - i*j}], i - 1], {j, 1, k/i}]]];
A[n_, k_] := Function [l, b[l[[1]], l[[2]], l[[2]]]][Sort[{n, k}]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jun 07 2021, after Alois P. Heinz *)

O.g.f. Product_{j >= 1} (1 + x^j/(1 - x^j) + y^j/(1 - y^j)) = Sum_{n,k >= 0} T(n,k)*x^n*y^k (see Wilf, Example 7).

Antidiagonal sums are A015128.

A382976 Expansion of Product_{k>=1} (1 + (2^k + 1) * x^k).

Original entry on oeis.org

1, 3, 5, 24, 44, 129, 384, 897, 2220, 5706, 15268, 35178, 89829, 212982, 526222, 1294263, 3087570, 7300896, 17726100, 41705904, 98782950, 236059794, 551697495, 1293417672, 3033232130, 7081297146, 16430673765, 38347412562, 88762751808, 204970377366, 473719894598

Offset: 0

Seiichi Manyama, Apr 11 2025

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -(2^n + 1).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Seiichi Manyama, Generalized Euler transform.

Cf. A322199, A382977, A382978.
Cf. A000051, A266964, A284593.
Cf. A079555.

n=30; CoefficientList[Normal@Series[Product[1+(2^k+1) x^k,{k,1,n}],{x,0,n}],x] (* Vincenzo Librandi, Apr 11 2025 *)

f(n) = -1;
g(n) = -(2^n+1);
a_vector(n) = my(b=vector(n, k, sumdiv(k, d, d*f(d)*g(d)^(k/d))), v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, b[j]*v[i-j+1])/i); v;

a(n) = Sum_{k=0..n} 2^k * A284593(k,n-k).

a(n) ~ A079555 * 2^(n-1) * exp(Pi*sqrt(n/3)) / (3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Apr 11 2025

A382955 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n * y^k] Product_{p prime} (1 + x^p + y^p).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 2, 0, 2, 1, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 1, 0, 2, 0, 2, 0, 1, 1, 0, 2

Offset: 0

Seiichi Manyama, Apr 10 2025

			Square array begins:
  1, 0, 1, 1, 0, 2, 0, 2, 1, 1, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  1, 0, 0, 1, 0, 1, 0, 1, 1, 0, ...
  1, 0, 1, 0, 0, 1, 0, 2, 0, 1, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  2, 0, 1, 1, 0, 2, 0, 2, 0, 1, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  2, 0, 1, 2, 0, 2, 0, 2, 1, 0, ...
  1, 0, 1, 0, 0, 0, 0, 1, 0, 1, ...
  1, 0, 0, 1, 0, 1, 0, 0, 1, 0, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..1 give A000586, A000004.
Main diagonal gives 2*A382871(n) (for n > 0).
Cf. A284593, A382956.

A(n,k) = A(k,n).

A382975 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n * y^k] Product_{j>=1} (1 + x^j - y^j).

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 0, -1, -1, 2, 0, -1, 0, -1, 2, 1, -1, -1, -1, -1, 3, 0, 0, 0, 0, -2, -2, 4, 1, 0, 0, -2, 0, -2, -2, 5, 0, 1, 0, -1, 0, -1, -2, -3, 6, 0, 1, 1, 0, -1, -1, -2, -3, -3, 8, 0, 1, 0, 0, -1, 2, -1, -2, -4, -5, 10, 0, 1, 1, 0, 1, -2, 0, -1, -2, -5, -5, 12

Offset: 0

Seiichi Manyama, Apr 11 2025

			Square array begins:
  1, -1, -1,  0,  0,  1,  0,  1, ...
  1,  0, -1, -1, -1,  0,  0,  1, ...
  1, -1,  0, -1,  0,  0,  0,  1, ...
  2, -1, -1,  0, -2, -1,  0,  0, ...
  2, -1, -2,  0,  0, -1, -1,  1, ...
  3, -2, -2, -1, -1,  2, -2,  0, ...
  4, -2, -2, -2, -1,  0,  0, -1, ...
  5, -3, -3, -2, -1,  0, -1,  4, ...

Columns k=0..2 give A000009, (-1)*A025147, (-1)*A015744.
Rows n=0..2 give A010815, A078616, A297054.
Main diagonal gives A382980.
Antidiagonal sums give A000007.
Cf. A284593.

cp's OEIS Frontend

A032302 G.f.: Product_{k>=1} (1 + 2*x^k).

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A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.

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Mathematica

A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

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A322210 G.f.: P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)), where P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k, as a square table of coefficients T(n,k) read by antidiagonals.

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A284592 Square array read by antidiagonals: T(n,k) is the number of pairs of partitions of n and k respectively, such that the pair of partitions have no part in common.

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A382976 Expansion of Product_{k>=1} (1 + (2^k + 1) * x^k).

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A382955 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n * y^k] Product_{p prime} (1 + x^p + y^p).

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