A325244 -id:A325244 - cp's OEIS frontend

A325242 Irregular triangle read by rows with zeros removed where T(n,k) is the number of integer partitions of n with k distinct multiplicities, n > 0.

1, 2, 3, 4, 1, 4, 3, 8, 3, 6, 9, 10, 12, 11, 19, 15, 26, 1, 13, 39, 4, 25, 47, 5, 19, 70, 12, 29, 89, 17, 33, 115, 28, 42, 148, 41, 39, 189, 69, 62, 235, 88, 55, 294, 141, 81, 362, 183, 1, 84, 450, 253, 5, 103, 558, 333, 8, 105, 669, 464, 17, 153, 817, 576, 29

Offset: 1

Gus Wiseman, Apr 15 2019

For example, the partition (32111) has multiplicities {1,1,3}, of which 2 are distinct, so is counted under T(8,2).

			Triangle begins:
   1
   2
   3
   4   1
   4   3
   8   3
   6   9
  10  12
  11  19
  15  26   1
  13  39   4
  25  47   5
  19  70  12
  29  89  17
  33 115  28
  42 148  41
  39 189  69
  62 235  88
  55 294 141
  81 362 183   1
Row n = 8 counts the following partitions:
  (8)         (332)
  (44)        (422)
  (53)        (611)
  (62)        (3221)
  (71)        (4211)
  (431)       (5111)
  (521)       (22211)
  (2222)      (32111)
  (3311)      (41111)
  (11111111)  (221111)
              (311111)
              (2111111)

Alois P. Heinz, Rows n = 1..200, flattened

Row lengths are A056556. Row sums are A000041. Column k = 1 is A047966. Column k = 2 is A325243.

Cf. A008284, A062770, A071625, A098859, A116608, A127002, A183558, A243978, A244515, A325244, A325268.

DeleteCases[Table[Length[Select[IntegerPartitions[n],Length[Union[Length/@Split[#]]]==k&]],{n,20},{k,n}],0,2]

A183558 Number of partitions of n containing a clique of size 1.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 7, 13, 16, 25, 33, 49, 61, 90, 113, 156, 198, 269, 334, 448, 556, 726, 902, 1163, 1428, 1827, 2237, 2817, 3443, 4302, 5219, 6478, 7833, 9632, 11616, 14197, 17031, 20712, 24769, 29925, 35688, 42920, 50980, 61059, 72318, 86206, 101837, 120941

Offset: 0

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(5) = 6, because 6 partitions of 5 contain (at least) one clique of size 1: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(1) = 1 through a(8) = 16 partitions are the following. The Heinz numbers of these partitions are given by A052485 (weak numbers).
  (1)  (2)  (3)   (4)    (5)     (6)      (7)       (8)
            (21)  (31)   (32)    (42)     (43)      (53)
                  (211)  (41)    (51)     (52)      (62)
                         (221)   (321)    (61)      (71)
                         (311)   (411)    (322)     (332)
                         (2111)  (3111)   (331)     (422)
                                 (21111)  (421)     (431)
                                          (511)     (521)
                                          (2221)    (611)
                                          (3211)    (3221)
                                          (4111)    (4211)
                                          (31111)   (5111)
                                          (211111)  (32111)
                                                    (41111)
                                                    (311111)
                                                    (2111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Column k=1 of A183568.
Cf. A000041, A007690, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.
Cf. A052485, A090858, A117571, A127002, A325241, A325242, A325244.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=1, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);

max = 50; f = (1 - Product[1 - x^j + x^(2*j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; CoefficientList[s, x] (* Jean-François Alcover, Oct 01 2014. Edited by Gus Wiseman, Apr 19 2019 *)

G.f.: (1-Product_{j>0} (1-x^(j)+x^(2*j))) / (Product_{j>0} (1-x^j)).
From Vaclav Kotesovec, Nov 15 2016: (Start)
a(n) = A000041(n) - A007690(n).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). (End)

a(0)=0 prepended by Gus Wiseman, Apr 19 2019

A090858 Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 2, 4, 6, 7, 8, 13, 15, 21, 25, 30, 39, 50, 58, 74, 89, 105, 129, 156, 185, 221, 264, 309, 366, 433, 505, 593, 696, 805, 941, 1090, 1258, 1458, 1684, 1933, 2225, 2555, 2922, 3346, 3823, 4349, 4961, 5644, 6402, 7267, 8234, 9309, 10525, 11886, 13393

Offset: 0

Vladeta Jovovic, Feb 12 2004

Number of solutions (p(1),p(2),...,p(n)), p(i)>=0,i=1..n, to p(1)+2*p(2)+...+n*p(n)=n such that |{i: p(i)<>0}| = p(1)+p(2)+...+p(n)-1.

Also number of partitions of n such that if k is the largest part, then, with exactly one exception, all the integers 1,2,...,k occur as parts. Example: a(7)=4 because we have [4,2,1], [3,3,1], [3,2,2] and [3,1,1,1,1]. - Emeric Deutsch, Apr 18 2006

			a(7) = 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(2) = 1 through a(11) = 13 partitions described in the name are the following (empty columns not shown). The Heinz numbers of these partitions are given by A060687.
  (11)  (22)   (221)  (33)   (322)   (44)    (441)   (55)    (443)
        (211)  (311)  (411)  (331)   (332)   (522)   (433)   (533)
                             (511)   (422)   (711)   (442)   (551)
                             (3211)  (611)   (3321)  (622)   (722)
                                     (3221)  (4221)  (811)   (911)
                                     (4211)  (4311)  (5221)  (4322)
                                             (5211)  (5311)  (4331)
                                                     (6211)  (4421)
                                                             (5411)
                                                             (6221)
                                                             (6311)
                                                             (7211)
                                                             (43211)
The a(2) = 1 through a(10) = 8 partitions described in Emeric Deutsch's comment are the following (empty columns not shown). The Heinz numbers of these partitions are given by A325284.
  (2)  (22)  (32)   (222)   (322)    (332)     (432)      (3322)
       (31)  (311)  (3111)  (331)    (431)     (3222)     (3331)
                            (421)    (2222)    (4221)     (22222)
                            (31111)  (3311)    (4311)     (42211)
                                     (4211)    (33111)    (43111)
                                     (311111)  (42111)    (331111)
                                               (3111111)  (421111)
                                                          (31111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A047967, A265251.
Column k=2 of A266477.
Cf. A008284, A046660, A060687, A116608, A117571, A127002, A325241, A325244.

g:=sum(x^(k*(k+1)/2)*((1-x^k)/x^(k-1)/(1-x)-k)/product(1-x^i,i=1..k),k=1..15): gser:=series(g,x=0,64): seq(coeff(gser,x,n),n=1..54); # Emeric Deutsch, Apr 18 2006
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n>i*(i+3-2*t)/2, 0,
     `if`(n=0, t, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i-1, t)+
     `if`(t=1 or 2*i>n, 0, b(n-2*i, i-1, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2015

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 3 - 2*t)/2, 0, If[n == 0, t, b[n, i - 1, t] + If[i > n, 0,  b[n - i, i - 1, t] + If[t == 1 || 2*i > n, 0, b[n - 2*i, i - 1, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[#]-Length[Union[#]]==1&]],{n,0,30}] (* Gus Wiseman, Apr 19 2019 *)

alist(n)=concat([0,0],Vec(sum(k=1,n\2,(x^(2*k)+x*O(x^n))/(1+x^k)*prod(j=1,n-2*k,1+x^j+x*O(x^n))))) \\ Franklin T. Adams-Watters, Nov 02 2015

G.f.: Sum_{k>0} x^(2*k)/(1+x^k) * Product_{k>0} (1+x^k). Convolution of 1-A048272(n) and A000009(n). a(n) = A036469(n) - A015723(n).
G.f.: sum(x^(k(k+1)/2)[(1-x^k)/x^(k-1)/(1-x)-k]/product(1-x^i,i=1..k), k=1..infinity). - Emeric Deutsch, Apr 18 2006
a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = 3^(1/4) * (1 - log(2)) / (2*Pi) = 0.064273294789... - Vaclav Kotesovec, May 24 2018

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004

a(0) added by Franklin T. Adams-Watters, Nov 02 2015

A117571 Expansion of (1+2x^2)/((1-x)(1-x^3)).

Original entry on oeis.org

1, 1, 3, 4, 4, 6, 7, 7, 9, 10, 10, 12, 13, 13, 15, 16, 16, 18, 19, 19, 21, 22, 22, 24, 25, 25, 27, 28, 28, 30, 31, 31, 33, 34, 34, 36, 37, 37, 39, 40, 40, 42, 43, 43, 45, 46, 46, 48, 49, 49, 51, 52, 52, 54, 55, 55, 57, 58, 58, 60, 61, 61, 63, 64, 64, 66, 67, 67, 69, 70, 70, 72

Offset: 0

Paul Barry, Mar 29 2006

Row sums of A116948.
Place n+2 equally-spaced points around a circle, labeled 0,1,2,...,n+1. For each i = 0..n+1 such that 2i != i mod n+2, draw an (undirected) chord from i to 2i mod n+2. Then a(n) is the number of distinct chords. - Kival Ngaokrajang, May 13 2016 (Edited by N. J. A. Sloane, Jun 23 2016)
From Gus Wiseman, Apr 19 2019: (Start)
Also the number of integer partitions of n + 2 with 1 fewer distinct multiplicities than (not necessarily distinct) parts. These are partitions of the form (x,x), (x,y), (x,x,y), or (x,y,y). For example, the a(0) = 1 through a(8) = 9 partitions are the following. The Heinz numbers of these partitions are given by A325270.
  (11)  (21)  (22)   (32)   (33)   (43)   (44)   (54)   (55)
              (31)   (41)   (42)   (52)   (53)   (63)   (64)
              (211)  (221)  (51)   (61)   (62)   (72)   (73)
                     (311)  (411)  (322)  (71)   (81)   (82)
                                   (331)  (332)  (441)  (91)
                                   (511)  (422)  (522)  (433)
                                          (611)  (711)  (442)
                                                        (622)
                                                        (811)
(End)

Kival Ngaokrajang, Illustration of initial terms
Burkard Polster, Times Tables, Mandelbrot and the Heart of Mathematics, Mathologer video (2015).
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A001045, A116948, A273724.

Cf. A090858, A127002, A323055, A325242, A325243, A325244, A325270.

[1 + Floor(2*n/3) + Floor((n+1)/3) : n in [0..100]]; // Wesley Ivan Hurt, Jul 25 2016

A117571:=n->1 + floor(2*n/3) + floor((n+1)/3): seq(A117571(n), n=0..100); # Wesley Ivan Hurt, Jul 25 2016

CoefficientList[Series[(1 + 2 x^2)/((1 - x) (1 - x^3)), {x, 0, 71}], x] (* Michael De Vlieger, May 13 2016 *)

G.f.: (1+2*x^2)/((1-x)*(1-x^3)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>3.
a(n) = cos(2*Pi*n/3+Pi/6)/sqrt(3)-sin(2*Pi*n/3+Pi/6)/3+(3n+2)/3.
a(n) = Sum_{k=0..n} 2*A001045(L((n-k+2)/3)) where L(j/p) is the Legendre symbol of j and p.
a(n) = 1 + floor((n+1)/3) + floor(2*n/3). - Wesley Ivan Hurt, Jul 25 2016
a(n) = n+sign((n-1) mod 3). - Wesley Ivan Hurt, Sep 25 2017

A127002 Number of partitions of n that have the form a+a+b+c where a,b,c are distinct.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 4, 3, 7, 8, 11, 11, 17, 17, 23, 23, 30, 31, 39, 38, 48, 49, 58, 58, 70, 70, 82, 82, 95, 96, 110, 109, 125, 126, 141, 141, 159, 159, 177, 177, 196, 197, 217, 216, 238, 239, 260, 260, 284, 284, 308, 308, 333, 334, 360, 359, 387, 388, 415, 415, 445

Offset: 1

Clark Kimberling, Jan 01 2007

From Gus Wiseman, Apr 19 2019: (Start)
Also the number of integer partitions of n - 4 of the form a+b, a+a+b, or a+a+b+c, ignoring ordering. A bijection can be constructed from the partitions described in the name by subtracting one from all parts and deleting zeros. These are also partitions with adjusted frequency depth (A323014, A325280) equal to their length plus one, and their Heinz numbers are given by A325281. For example, the a(7) = 1 through a(13) = 11 partitions are:
  (21)  (31)   (32)   (42)   (43)    (53)    (54)
        (211)  (41)   (51)   (52)    (62)    (63)
               (221)  (411)  (61)    (71)    (72)
               (311)         (322)   (332)   (81)
                             (331)   (422)   (441)
                             (511)   (611)   (522)
                             (3211)  (3221)  (711)
                                     (4211)  (3321)
                                             (4221)
                                             (4311)
                                             (5211)
(End)

			a(10) counts these partitions: {1,1,2,6}, (1,1,3,5), {2,2,1,5}.
a(11) counts {1,1,2,7}, {1,1,3,6}, {1,1,4,5}, {2,2,1,6}, {2,2,3,4}, {3,3,1,4}, {4,4,1,2}
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(7) = 1 through a(13) = 11 partitions of the form a+a+b+c are the following. The Heinz numbers of these partitions are given by A085987.
  (3211)  (3221)  (3321)  (5221)  (4322)  (4332)  (4432)
          (4211)  (4221)  (5311)  (4331)  (4431)  (5332)
                  (4311)  (6211)  (4421)  (5322)  (5422)
                  (5211)          (5411)  (5331)  (5521)
                                  (6221)  (6411)  (6322)
                                  (6311)  (7221)  (6331)
                                  (7211)  (7311)  (6511)
                                          (8211)  (7411)
                                                  (8221)
                                                  (8311)
                                                  (9211)
(End)

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

Cf. A000041, A008284, A085987, A090858, A116608, A117571, A183558, A325242, A325244, A325280, A325281.

R:=PowerSeriesRing(Integers(), 70); [0,0,0,0,0,0] cat Coefficients(R!( x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) )); // G. C. Greubel, May 30 2019

g:=sum(sum(sum(x^(i+j+k)*(x^i+x^j+x^k),i=1..j-1),j=2..k-1),k=3..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..65); # Emeric Deutsch, Jan 05 2007
isA127002 := proc(p) local s; if nops(p) = 4 then s := convert(p,set) ; if nops(s) = 3 then RETURN(1) ; else RETURN(0) ; fi ; else RETURN(0) ; fi ; end:
A127002 := proc(n) local part,res,p; part := combinat[partition](n) ; res := 0 ; for p from 1 to nops(part) do res := res+isA127002(op(p,part)) ; od ; RETURN(res) ; end:
for n from 1 to 200 do print(A127002(n)) ; od ; # R. J. Mathar, Jan 07 2007

Table[Length[Select[IntegerPartitions[n],Sort[Length/@Split[#]]=={1,1,2}&]],{n,70}] (* Gus Wiseman, Apr 19 2019 *)
Rest[CoefficientList[Series[x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)), {x,0,70}], x]] (* G. C. Greubel, May 30 2019 *)

my(x='x+O('x^70)); concat(vector(6), Vec(x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)))) \\ G. C. Greubel, May 30 2019

a=(x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4))).series(x, 70).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 30 2019

G.f.: x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) - Vladeta Jovovic, Jan 03 2007
G.f.: Sum_{k>=3} Sum_{j=2..k-1} Sum_{m=1..j-1} x^(m+j+k)*(x^m +x^j +x^k). - Emeric Deutsch, Jan 05 2007
a(n) = binomial(floor((n-1)/2),2) - floor((n-1)/3) - floor((n-1)/4) + floor(n/4). - Mircea Merca, Nov 23 2013
a(n) = A005044(n-4) + 2*A005044(n-3) + 3*A005044(n-2). - R. J. Mathar, Nov 23 2013

A378622 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 2, 0, -1, -2, -3, 3, 1, 1, 2, 4, 7, 4, 1, 0, -1, -3, -7, -14, 5, 1, 0, 0, 1, 4, 11, 25, 6, 1, 0, 0, 0, -1, -5, -16, -41, 8, 2, 1, 1, 1, 1, 2, 7, 23, 64, 10, 2, 0, -1, -2, -3, -4, -6, -13, -36, -100, 12, 2, 0, 0, 1, 3, 6, 10, 16, 29, 65, 165

Offset: 0

Gus Wiseman, Dec 13 2024

			As a table (read by antidiagonals downward):
        n=0:  n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:
  ----------------------------------------------------------
  k=0:   1     1     1     2     2     3     4     5     6
  k=1:   0     0     1     0     1     1     1     1     2
  k=2:   0     1    -1     1     0     0     0     1     0
  k=3:   1    -2     2    -1     0     0     1    -1     0
  k=4:  -3     4    -3     1     0     1    -2     1     1
  k=5:   7    -7     4    -1     1    -3     3     0    -3
  k=6: -14    11    -5     2    -4     6    -3    -3     7
  k=7:  25   -16     7    -6    10    -9     0    10   -14
  k=8: -41    23   -13    16   -19     9    10   -24    24
  k=9:  64   -36    29   -35    28     1   -34    48   -34
As a triangle (read by rows):
   1
   1   0
   1   0   0
   2   1   1   1
   2   0  -1  -2  -3
   3   1   1   2   4   7
   4   1   0  -1  -3  -7 -14
   5   1   0   0   1   4  11  25
   6   1   0   0   0  -1  -5 -16 -41
   8   2   1   1   1   1   2   7  23  64

Rows are: A000009 (k=0), A087897 (k=1, without first term), A378972 (k=2).
For primes we have A095195 or A376682.
For partitions we have A175804.
First column is A293467 (up to sign).
For composites we have A377033.
For squarefree numbers we have A377038.
For nonsquarefree numbers we have A377046.
For prime powers we have A377051.
Position of first zero in each row is A377285.
Triangle's row-sums are A378970, absolute A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A325242, A225485 or A325280.

nn=20;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A378972 Second differences of the strict partition numbers A000009.

Original entry on oeis.org

0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, 3, 2, 3, 4, 3, 4, 6, 4, 6, 8, 6, 9, 10, 9, 12, 14, 13, 16, 19, 18, 22, 26, 24, 30, 34, 34, 40, 45, 46, 53, 60, 62, 70, 79, 82, 93, 104, 108, 122, 136, 142, 160, 176, 186, 208, 228, 243, 268

Offset: 0

Gus Wiseman, Dec 14 2024

sign

			The strict partition numbers begin (A000009):
  1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, ...
with differences (A087897 without first term):
  0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, ...
with differences (a(n)):
  0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, ...

For primes we have A036263.
The version for partitions is A053445.
For composites we have A073445.
For squarefree numbers we have A376590.
For nonsquarefree numbers we have A376593.
For powers of primes (inclusive) we have A376596.
For non powers of primes (inclusive) we have A376599.
Second row of A378622. See also:
- A293467 gives first column (up to sign).
- A377285 gives position of first zero in each row.
- A378970 gives row-sums.
- A378971 gives absolute value row-sums.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A225485, A325242, A325268.

Differences[Table[PartitionsQ[n],{n,0,100}],2]

A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

Original entry on oeis.org

0, 1, 1, 5, 5, 8, 20, 7, 22

Offset: 0

Gus Wiseman, Dec 12 2024

Open problem: Do the 9th differences of the strict integer partition numbers contain a zero? If so, we must have a(9) > 10^5.

a(12) = 47. Conjecture: a(n) = 0 for n > 12. - Chai Wah Wu, Dec 15 2024

			The 7th differences of A000009 are: 25, -16, 7, -6, 10, -9, 0, 10, ... so a(7) = 7.

For primes we have A376678.
For composites we have A377037.
For squarefree numbers we have A377042.
For nonsquarefree numbers we have A377050.
For prime-powers we have A377055.
Position of first zero in each row of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives first column (up to sign).
- A378970 gives row-sums.
- A378971 gives row-sums of absolute value.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A325242, A325268, A225485 or A325280.

Table[Position[Differences[PartitionsQ/@Range[0,100],k],0][[1,1]],{k,1,8}]

a(n, nn=100) = my(q='q+O('q^nn), v=Vec(eta(q^2)/eta(q))); for (i=1, n, my(w=vector(#v-1, k, v[k+1]-v[k])); v = w;); my(vz=select(x->x==0, v, 1)); if (#vz, vz[1]); \\ Michel Marcus, Dec 15 2024

A378970 Antidiagonal-sums of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

Original entry on oeis.org

1, 1, 1, 5, -4, 18, -20, 47, -56, 110, -153, 309, -532, 1045, -1768, 2855, -3620, 2928, 2927, -20371, 62261, -148774, 314112, -613835, 1155936, -2175658, 4244218, -8753316, 19006746, -42471491, 95234915, -210395017, 453414314, -949507878, 1931940045

Offset: 0

Gus Wiseman, Dec 14 2024

sign

			Antidiagonal 4 of A378622 is (2, 0, -1, -2, -3), so a(4) = -4.

For primes we have A140119 or A376683, absolute value A376681 or A376684.
For composites we have A377034, absolute value A377035.
For squarefree numbers we have A377039, absolute value A377040.
For nonsquarefree numbers we have A377047, absolute value A377048.
For prime powers we have A377052, absolute value A377053.
For partition numbers we have A377056, absolute value A378621.
Row-sums of the triangular form of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives the first column (up to sign).
- A377285 gives position of first zero in each row.
The unsigned version is A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A325244.

nn=30;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A378971 Antidiagonal-sums of absolute value of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

Original entry on oeis.org

1, 1, 1, 5, 8, 18, 30, 47, 70, 110, 177, 309, 574, 1063, 1892, 3107, 4598, 6166, 8737, 20603, 62457, 149132, 314116, 614093, 1155968, 2176048, 4244322, 8753864, 19006756, 42472117, 95235017, 210396059, 453414950, 949510166, 1931941261, 3826650257, 7400745917

Offset: 0

Gus Wiseman, Dec 14 2024

nonn

			Antidiagonal 4 of A378622 is (2, 0, -1, -2, -3), so a(4) = 8.

For primes we have A376681 or A376684, signed version A140119 or A376683.
For composites we have A377035, signed version A377034.
For squarefree numbers we have A377040, signed version A377039.
For nonsquarefree numbers we have A377048, signed version A377049.
For prime powers we have A377053, signed version A377052.
For partition numbers we have A378621, signed version A377056.
Row-sums of the triangular form of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives the first column (up to sign).
- A377285 gives position of first zero in each row.
The signed version is A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A325244, A325245, A325268.

nn=30;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

cp's OEIS Frontend

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