A087897 -id:A087897 - cp's OEIS frontend

A378972 Second differences of the strict partition numbers A000009.

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]

A377056 Antidiagonal-sums of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 3, 11, 2, 36, -27, 142, -207, 595, -1066, 2497, -4878, 10726, -22189, 48383, -103318, 224296, -480761, 1030299, -2186942, 4626313, -9740648, 20492711, -43109372, 90843475, -191769296, 405528200, -858373221, 1817311451, -3845483855, 8129033837

Offset: 0

Gus Wiseman, Dec 12 2024

sign

			Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 3.

For primes we have A140119 or A376683, unsigned A376681 or A376684.
These are the antidiagonal-sums of A175804.
First column of the same array is A281425.
For composites we have A377034, unsigned A377035.
For squarefree numbers we have A377039, unsigned A377040.
For nonsquarefree numbers we have A377049, unsigned A377048.
For prime powers we have A377052, unsigned A377053.
The unsigned version is A378621.
The version for strict partitions is A378970 (row-sums of A378622), unsigned A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A075526, A116608, A225486, A293467, A325242, A325245, A325268, A377285.

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

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}]

A323054 Number of strict integer partitions of n with no 1's such that no part is a power of any other part.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 3, 4, 4, 6, 6, 8, 9, 12, 13, 16, 19, 21, 25, 30, 36, 40, 47, 53, 63, 71, 83, 94, 107, 121, 140, 159, 180, 204, 233, 260, 296, 334, 377, 421, 474, 532, 598, 668, 750, 835, 933, 1038, 1163, 1292, 1435, 1597, 1771, 1966, 2180, 2421, 2673

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			The a(2) = 1 through a(13) = 8 strict integer partitions (A = 10, B = 11, C = 12, D = 13):
  (2)  (3)  (4)  (5)   (6)  (7)   (8)   (9)   (A)    (B)    (C)    (D)
                 (32)       (43)  (53)  (54)  (64)   (65)   (75)   (76)
                            (52)  (62)  (63)  (73)   (74)   (84)   (85)
                                        (72)  (532)  (83)   (A2)   (94)
                                                     (92)   (543)  (A3)
                                                     (632)  (732)  (B2)
                                                                   (643)
                                                                   (652)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

Cf. A001597, A007916, A025147, A052410, A078374, A087897, A120641, A275972, A303362, A323053, A323087, A323088.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],UnsameQ@@#,stableQ[#,IntegerQ[Log[#1,#2]]&]]&]],{n,30}]

A323094 Number of strict integer partitions of n where no part is 2^k times any other part, for any k > 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 4, 4, 4, 5, 7, 8, 10, 12, 12, 15, 17, 20, 24, 27, 33, 35, 41, 48, 54, 61, 69, 79, 87, 101, 113, 128, 144, 159, 181, 201, 225, 251, 281, 311, 347, 388, 428, 477, 525, 579, 643, 712, 788, 868, 954, 1051, 1155, 1272, 1398, 1534, 1682, 1840, 2016

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			The a(1) = 1 through a(12) = 8 strict integer partitions (A = 10, B = 11, C = 12):
  (1)  (2)  (3)  (4)   (5)   (6)   (7)   (8)   (9)    (A)    (B)    (C)
                 (31)  (32)  (51)  (43)  (53)  (54)   (64)   (65)   (75)
                                   (52)  (62)  (72)   (73)   (74)   (93)
                                   (61)  (71)  (531)  (91)   (83)   (A2)
                                                      (532)  (92)   (B1)
                                                             (A1)   (543)
                                                             (731)  (651)
                                                                    (732)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400

Cf. A018819, A087897, A120641, A276431, A303362, A323054, A323092, A323093.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&stableQ[#,IntegerQ[Log[2,#1/#2]]&]&]],{n,30}]

A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 1, 2, 2, 1, 1, 0, 5, 1, 3, 2, 2, 1, 1, 0, 6, 2, 3, 4, 3, 2, 1, 1, 0, 8, 2, 4, 5, 4, 3, 2, 1, 1, 0, 10, 2, 5, 7, 6, 5, 3, 2, 1, 1, 0, 12, 3, 6, 8, 9, 6, 5, 3, 2, 1, 1, 0, 15, 3, 8, 11, 11, 10, 7, 5, 3, 2, 1, 1, 0

Offset: 0

Gus Wiseman, Apr 20 2021

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.

			Triangle begins:
   1
   1   0
   1   1   0
   2   0   1   0
   2   1   1   1   0
   3   1   1   1   1   0
   4   1   2   2   1   1   0
   5   1   3   2   2   1   1   0
   6   2   3   4   3   2   1   1   0
   8   2   4   5   4   3   2   1   1   0
  10   2   5   7   6   5   3   2   1   1   0
  12   3   6   8   9   6   5   3   2   1   1   0
  15   3   8  11  11  10   7   5   3   2   1   1   0
  18   4   9  13  15  13  10   7   5   3   2   1   1   0
  22   5  10  17  19  18  14  11   7   5   3   2   1   1   0
  27   5  13  20  24  23  20  14  11   7   5   3   2   1   1   0
For example, row n = 9 counts the following partitions:
  (3321)       (432)   (333)      (54)      (522)    (63)    (72)   (81)  (9)
  (22221)      (3222)  (4311)     (441)     (531)    (621)   (711)
  (32211)              (33111)    (4221)    (5211)   (6111)
  (222111)             (3111111)  (42111)   (51111)
  (321111)                        (411111)
  (2211111)
  (21111111)
  (111111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Column k = 0 is A000009.
Row sums are A000041.
Central diagonal is A000041.
Column k = 1 is A087897.
The version for least gap is A264401, with Heinz number encoding A257993.
The version for greatest difference is A286469 or A286470.
An encoding (of greatest gap) using Heinz numbers is A339662.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A048004 counts compositions by greatest part.
A056239 adds up prime indices, row sums of A112798.
A064391 is the version for crank.
A064428 counts partitions of nonnegative crank.
A073491 list numbers with gap-free prime indices.
A107428 counts gap-free compositions.
A238709/A238710 counts partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A002110, A018818, A063250, A088860, A098743, A279945.

maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[Length[Select[IntegerPartitions[n],maxgap[#]==k&]],{n,0,15},{k,0,n}]

S(n,k)={if(k>n, O(x*x^n), x^k*(S(n-k,k+1) + 1)/(1 - x^k))}
ColGf(k,n) = {(k==0) + S(n,k+1)/prod(j=1, k-1, 1 - x^j + O(x^max(1,n-k)))}
A(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
{ my(M=A(10)); for(i=1, #M, print(M[i,1..i])) } \\ Andrew Howroyd, Jan 13 2024

Offset corrected by Andrew Howroyd, Jan 13 2024

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}]

A319613 a(n) = prime(n) * prime(2n).

Original entry on oeis.org

6, 21, 65, 133, 319, 481, 731, 1007, 1403, 2059, 2449, 3293, 4141, 4601, 5311, 6943, 8201, 9211, 10921, 12283, 13213, 15247, 16517, 19847, 22213, 24139, 25853, 28141, 29539, 31753, 37211, 40741, 43429, 46843, 52001, 54209, 58561, 62429, 66299, 70757, 75359

Offset: 1

Gus Wiseman, Jan 07 2019

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A018819, A031215, A031368, A056239, A087897, A101417, A112798, A120641, A320340, A323092, A323093, A323094.

a:= n-> (p-> p(n)*p(2*n))(ithprime):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 08 2019

Mathematica
```
Table[Prime[n]*Prime[2*n],{n,50}]
```

a(n) = prime(n)*prime(2*n) \\ Felix Fröhlich, Jan 09 2019

A320591 Expansion of Product_{k>=1} (1 + x^k/(1 + x)^k).

Original entry on oeis.org

1, 1, 0, 1, -2, 4, -7, 11, -16, 23, -36, 65, -129, 256, -473, 772, -1028, 835, 776, -5755, 17562, -41750, 86678, -165145, 299949, -541837, 1020029, -2068203, 4509512, -10252952, 23465297, -52762788, 115160832, -243018459, 496094524, -982431070, 1894710043, -3574095362

Offset: 0

Ilya Gutkovskiy, Oct 16 2018

After the first term, this is the second term of the n-th differences of A000009, or column n=1 of A378622. - Gus Wiseman, Feb 03 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000593, A320589, A307548.
The version for non-strict partitions is A320590, row n=1 of A175804.
Column n=1 (except first term) of A378622. See also A293467, A377285, A378970, A378971, A380412 (column n=0).
A000009 counts strict integer partitions, differences A087897, A378972.
A266232 gives zero-based binomial transform of strict partitions, differences A129519.
Cf. A000041, A007442, A002865, A008284, A095195, A103446, A116608, A218482, A281425, A320568.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1 + x^k/(1 + x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018

seq(coeff(series(mul((1+x^k/(1+x)^k),k=1..n),x,n+1), x, n), n = 0 .. 37); # Muniru A Asiru, Oct 16 2018

nmax = 37; CoefficientList[Series[Product[(1 + x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]
Prepend[Table[Differences[PartitionsQ/@Range[0,k+1],k][[2]],{k,0,30}],1] (* Gus Wiseman, Jan 29 2025 *)

m=50; x='x+O('x^m); Vec(prod(k=1, m+2, (1 + x^k/(1 + x)^k))) \\ G. C. Greubel, Oct 29 2018

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*((1 + x)^k - x^k))).
G.f.: exp(Sum_{k>=1} A000593(k)*x^k/(k*(1 + x)^k)).
From Peter Bala, Dec 22 2020: (Start)
O.g.f.: Sum_{n >= 0}  x^(n*(n+1)/2)/Product_{k = 1..n} ((1 + x)^k - x^k). Cf. A307548.
Conjectural o.g.f.: (1/2) * Sum_{n >= 0} x^(n*(n-1)/2)*(1 + x)^n/( Product_{k = 1..n} ( (1 + x)^k - x^k ) ). (End)
a(n+1) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k) A000009(k+1). - Gus Wiseman, Feb 03 2025

cp's OEIS Frontend

A378972 Second differences of the strict partition numbers A000009.

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A377056 Antidiagonal-sums of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

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

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

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A323054 Number of strict integer partitions of n with no 1's such that no part is a power of any other part.

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A323094 Number of strict integer partitions of n where no part is 2^k times any other part, for any k > 0.

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A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

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

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A319613 a(n) = prime(n) * prime(2n).

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