cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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).

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

Views

Author

Gus Wiseman, Dec 14 2024

Keywords

Examples

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

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Dec 14 2024

Keywords

Examples

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

Crossrefs

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.

Programs

  • Mathematica
    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}]

A175804 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the n-th term of the k-th differences of partition numbers A000041.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, -1, 0, 1, 3, 2, 1, 1, 2, 5, -4, -2, -1, 0, 2, 7, 9, 5, 3, 2, 2, 4, 11, -21, -12, -7, -4, -2, 0, 4, 15, 49, 28, 16, 9, 5, 3, 3, 7, 22, -112, -63, -35, -19, -10, -5, -2, 1, 8, 30, 249, 137, 74, 39, 20, 10, 5, 3, 4, 12, 42, -539, -290, -153, -79, -40, -20, -10, -5, -2, 2, 14, 56
Offset: 0

Views

Author

Alois P. Heinz, Dec 04 2010

Keywords

Comments

Odlyzko showed that the k-th differences of A000041(n) alternate in sign with increasing n up to a certain index n_0(k) and then stay positive.
Are there any zeros after the first four, which all lie in columns k = 1, 2? - Gus Wiseman, Dec 15 2024

Examples

			Square array A(n,k) begins:
   1,  0,  1, -1,  2,  -4,   9,  ...
   1,  1,  0,  1, -2,   5, -12,  ...
   2,  1,  1, -1,  3,  -7,  16,  ...
   3,  2,  0,  2, -4,   9, -19,  ...
   5,  2,  2, -2,  5, -10,  20,  ...
   7,  4,  0,  3, -5,  10, -20,  ...
  11,  4,  3, -2,  5, -10,  22,  ...

Links

Crossrefs

Columns k=0-5 give: A000041, A002865, A053445, A072380, A081094, A081095.
Main diagonal gives A379378.
Cf. A119712, A155861.
For primes we have A095195 or A376682.
Row n = 0 is A281425.
Row n = 1 is A320590 except first term.
For composites we have A377033.
For squarefree numbers we have A377038.
For nonsquarefree numbers we have A377046.
For prime powers we have A377051.
Antidiagonal sums are A377056, absolute value version A378621.
The version for strict partitions is A378622, first column A293467.
A000009 counts strict integer partitions, differences A087897, A378972.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A225485 or A325280, A377285, A378970.

Programs

  • Maple
    A41:= combinat[numbpart]:
DD:= proc(p) proc(n) option remember; p(n+1) -p(n) end end:
A:= (n,k)-> (DD@@k)(A41)(n):
seq(seq(A(n, d-n), n=0..d), d=0..11);
  • Mathematica
    max = 11; a41 = Array[PartitionsP, max+1, 0]; a[n_, k_] := Differences[a41, k][[n+1]]; Table[a[n, k-n], {k, 0, max}, {n, 0, k}] // Flatten (* Jean-François Alcover, Aug 29 2014 *)
nn=5;Table[Table[Sum[(-1)^(k-i)*Binomial[k,i]*PartitionsP[n+i],{i,0,k}],{k,0,nn}],{n,0,nn}] (* Gus Wiseman, Dec 15 2024 *)

Formula

A(n,k) = (Delta^(k) A000041)(n).
A(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A000041(n+i). In words, row x is the inverse zero-based binomial transform of A000041 shifted left x times. - Gus Wiseman, Dec 15 2024

A281425 a(n) = [q^n] (1 - q)^n / Product_{j=1..n} (1 - q^j).

Original entry on oeis.org

1, 0, 1, -1, 2, -4, 9, -21, 49, -112, 249, -539, 1143, -2396, 5013, -10550, 22420, -48086, 103703, -223806, 481388, -1029507, 2187944, -4625058, 9742223, -20490753, 43111808, -90840465, 191773014, -405523635, 858378825, -1817304609, 3845492204, -8129023694, 17162802918, -36191083386
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 05 2017

Keywords

Comments

a(n) is n-th term of the Euler transform of -n + 1, 1, 1, 1, ...
Inverse zero-based binomial transform of A000041. The version for strict partitions is A380412, or A293467 up to sign. - Gus Wiseman, Feb 06 2025

Links

Crossrefs

Cf. A128566, A292463, A292541, A292613.
Cf. A000041, A002865, A053445, A275638, A275639, A275640, A275641, A275642, A275643, A275644.
Cf. A218481, A294466, A095051.
Cf. A266232, A294467, A293467, A294468.
Row n=0 of A175804 (row n=1 is A320590 except first term).
Cf. A000009, A008284, A087897, A116608, A129519, A225486, A320591, A377056, A378622.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0,
      combinat[numbpart](n), b(n, k-1)-b(n-1, k-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 21 2024
  • Mathematica
    Table[SeriesCoefficient[(1 - q)^n / Product[(1 - q^j), {j, 1, n}], {q, 0, n}], {n, 0, 35}]
Table[SeriesCoefficient[(1 - q)^n QPochhammer[q^(1 + n), q]/QPochhammer[q, q], {q, 0, n}], {n, 0, 35}]
Table[SeriesCoefficient[1/QFactorial[n, q], {q, 0, n}], {n, 0, 35}]
Table[Differences[PartitionsP[Range[0, n]], n], {n, 0, 35}] // Flatten
Table[Sum[(-1)^j*Binomial[n, j]*PartitionsP[n-j], {j, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 06 2017 *)

Formula

a(n) = [q^n] 1/((1 + q)*(1 + q + q^2)*...*(1 + q + ... + q^(n-1))).
a(n) = Sum_{j=0..n} (-1)^j * binomial(n, j) * A000041(n-j). - Vaclav Kotesovec, Oct 06 2017
a(n) ~ (-1)^n * 2^(n - 3/2) * exp(Pi*sqrt(n/12) + Pi^2/96) / (sqrt(3)*n). - Vaclav Kotesovec, May 07 2018

A293467 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * q(k), where q(k) is A000009 (partitions into distinct parts).

Original entry on oeis.org

1, 0, 0, -1, -3, -7, -14, -25, -41, -64, -100, -165, -294, -550, -1023, -1795, -2823, -3658, -2882, 2873, 20435, 62185, 148863, 314008, 613957, 1155794, 2175823, 4244026, 8753538, 19006490, 42471787, 95234575, 210395407, 453413866, 949508390, 1931939460
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 09 2017

Keywords

Comments

Multiply by (-1)^n to get A380412, which is the first term of the n-th differences of the strict partition numbers, or column n=0 of A378622. - Gus Wiseman, Feb 04 2025

Links

Crossrefs

Cf. A025147, A266232, A294467, A294468.
Cf. A218481, A294466, A095051.
The non-strict version is the absolute value of A281425; see A175804, A320590.
Up to sign, same as A380412. See A320591, A377285, A378970, A378971.
A000009 counts strict integer partitions, differences A087897.
Cf. A000041, A002865, A007442, A008284, A116608.

Programs

  • Mathematica
    Table[Sum[(-1)^k * Binomial[n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 50}]

A320590 Expansion of Product_{k>=1} 1/(1 - x^k/(1 + x)^k).

Original entry on oeis.org

1, 1, 1, 0, 1, -2, 5, -12, 28, -63, 137, -290, 604, -1253, 2617, -5537, 11870, -25666, 55617, -120103, 257582, -548119, 1158437, -2437114, 5117165, -10748530, 22621055, -47728657, 100932549, -213750621, 452855190, -958925784, 2028187595, -4283531490, 9033779224
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 16 2018

Keywords

Comments

The zero-based binomial transform of this sequence is A000070, and if we remove first terms it becomes A000041.

Links

Crossrefs

Cf. A000203, A103446, A218482, A320568.
Row n=1 of A175804 (except first term). Row n=0 is A281425.
The version for strict partitions is A320591, row n=1 of A378622, first column A293467.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865.
Cf. A047966, A008284, A377056, A378621.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1 - x^k/(1 + x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018
  • Maple
    seq(coeff(series(mul(1/(1-x^k/(1+x)^k),k=1..n),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 16 2018
  • Mathematica
    nmax = 34; CoefficientList[Series[Product[1/(1 - x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 34; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]
  • PARI
    m=50; x='x+O('x^m); Vec(prod(k=1, m+2, 1/(1 - x^k/(1 + x)^k))) \\ G. C. Greubel, Oct 29 2018

Formula

G.f.: exp(Sum_{k>=1} x^k/(k*((1 + x)^k - x^k))).
G.f.: exp(Sum_{k>=1} sigma(k)*x^k/(k*(1 + x)^k)).

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

Views

Author

Gus Wiseman, Dec 14 2024

Keywords

Examples

			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, ...

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Dec 12 2024

Keywords

Examples

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

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Dec 12 2024

Keywords

Comments

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

Examples

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

Crossrefs

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.

Programs

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

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

Views

Author

Ilya Gutkovskiy, Oct 16 2018

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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
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