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.

Showing 1-8 of 8 results.

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

A292508 Number A(n,k) of partitions of n with k kinds of 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 2, 1, 4, 7, 7, 5, 2, 1, 5, 11, 14, 12, 7, 4, 1, 6, 16, 25, 26, 19, 11, 4, 1, 7, 22, 41, 51, 45, 30, 15, 7, 1, 8, 29, 63, 92, 96, 75, 45, 22, 8, 1, 9, 37, 92, 155, 188, 171, 120, 67, 30, 12, 1, 10, 46, 129, 247, 343, 359, 291, 187, 97, 42, 14
Offset: 0

Views

Author

Alois P. Heinz, Sep 17 2017

Keywords

Comments

Partial sum operator applied to column k gives column k+1.
A(n,k) is also defined for k < 0. All given formulas and programs can be applied also if k is negative.

Examples

			Square array A(n,k) begins:
  1,  1,  1,   1,   1,    1,    1,    1,     1, ...
  0,  1,  2,   3,   4,    5,    6,    7,     8, ...
  1,  2,  4,   7,  11,   16,   22,   29,    37, ...
  1,  3,  7,  14,  25,   41,   63,   92,   129, ...
  2,  5, 12,  26,  51,   92,  155,  247,   376, ...
  2,  7, 19,  45,  96,  188,  343,  590,   966, ...
  4, 11, 30,  75, 171,  359,  702, 1292,  2258, ...
  4, 15, 45, 120, 291,  650, 1352, 2644,  4902, ...
  7, 22, 67, 187, 478, 1128, 2480, 5124, 10026, ...

Links

Crossrefs

Columns k=0-10 give: A002865, A000041, A000070, A014153, A014160, A014161, A120477, A320753, A320754, A320755, A320756.
Rows n=0-4 give: A000012, A001477, A000124, A004006(k+1), A027927(k+3).
Main diagonal gives A292463.
A(n,n+1) gives A292613.
Cf. A292622, A292741, A292745.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)+k-1)*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1, `if`(k<1,
      A(n, k+1)-A(n-1, k+1), `if`(k=1, combinat[numbpart](n),
      A(n-1, k)+A(n, k-1))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
# third Maple program:
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      binomial(k+n-1, n), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, Binomial[k + n - 1, n], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 17 2018, translated from 3rd Maple program *)

Formula

G.f. of column k: 1/(1-x)^k * 1/Product_{j>1} (1-x^j).
Column k is Euler transform of k,1,1,1,... .
For fixed k>=0, A(n,k) ~ 2^((k-5)/2) * 3^((k-2)/2) * n^((k-3)/2) * exp(Pi*sqrt(2*n/3)) / Pi^(k-1). - Vaclav Kotesovec, Oct 24 2018

A292503 Number of partitions of n with n sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 1, 3, 7, 20, 63, 233, 966, 4454, 22404, 121616, 706362, 4361977, 28494493, 196087988, 1416515642, 10709058487, 84505818259, 694397612486, 5929368380664, 52513737017847, 481577858196052, 4565851595293151, 44692014464166068, 451058715629365617
Offset: 0

Views

Author

Alois P. Heinz, Sep 17 2017

Keywords

Examples

			a(2) = 3: 2, 1a1a, 1a1b.
a(3) = 7: 3, 21a, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1a1b1c.

Links

Crossrefs

Cf. A292462, A292463, A292507.
Main diagonal of A292745.
Row sums of A292746.

Programs

  • Maple
    f:= (n, k)-> add(Stirling2(n, j), j=0..k):
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      f(n, k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..30);
  • Mathematica
    f[n_, k_] := Sum[StirlingS2[n, j], {j, 0, k}];
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, f[n, k], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

A292462 Number of partitions of n with n sorts of part 1.

Original entry on oeis.org

1, 1, 5, 31, 278, 3287, 48256, 843567, 17081639, 392869430, 10112244792, 287927207846, 8984122319997, 304828239096197, 11173376516829974, 439988449921648076, 18523908107054523591, 830292183207722271065, 39475390430795389762048, 1984220622132901208082220
Offset: 0

Views

Author

Alois P. Heinz, Sep 16 2017

Keywords

Examples

			a(2) = 5: 2, 1a1a, 1a1b, 1b1a, 1b1b.

Links

Crossrefs

Cf. A002865, A246935, A292463, A292503, A292507, A292567.
Main diagonal of A292741.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^n,
      `if`(i>n, 0, b(n-i, i, k))+b(n, i-1, k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..23);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^n, If[i > n, 0, b[n - i, i, k]] + b[n, i - 1, k]];
a[0] = 1; a[n_] := b[n, n, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 19 2018, translated from Maple *)

Formula

a(n) = [x^n] 1/(1-n*x) * Product_{j=2..n} 1/(1-x^j).
a(n) ~ n^n * (1 + 1/n^2 + 1/n^3 + 2/n^4 + 2/n^5 + 4/n^6 + 4/n^7 + 7/n^8 + 8/n^9 + 12/n^10), for coefficients see A002865. - Vaclav Kotesovec, Sep 19 2017
a(n) = Sum_{j=0..n} A002865(j) * n^(n-j). - Alois P. Heinz, Sep 22 2017

A292507 Number of partitions of n with up to n distinct kinds of 1.

Original entry on oeis.org

1, 1, 2, 5, 13, 33, 82, 201, 488, 1176, 2817, 6714, 15931, 37647, 88628, 207914, 486158, 1133304, 2634339, 6106953, 14121157, 32573842, 74968044, 172164086, 394561089, 902471184, 2060338222, 4695324425, 10681885697, 24261437446, 55017434305, 124573678280
Offset: 0

Views

Author

Alois P. Heinz, Sep 17 2017

Keywords

Examples

			a(3) = 5: 3, 21a, 21b, 21c, 1a1b1c.
a(4) = 13: 4, 31a, 31b, 31c, 31d, 22, 21a1b, 21a1c, 21a1d, 21b1c, 21b1d, 21c1d, 1a1b1c1d.

Links

Crossrefs

Main diagonal of A292622.
Cf. A292462, A292463, A292503.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0 or i=1,
      binomial(k, n), `if`(i>n, 0, b(n-i, i, k))+b(n, i-1, k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..35);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, Binomial[k, n], If[i > n, 0, b[n - i, i, k]] + b[n, i - 1, k]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 20 2018, translated from Maple *)

Formula

Conjecture: log(a(n)) ~ log(2)*n + Pi*sqrt(n/3) - 3*log(n)/2. - Vaclav Kotesovec, May 11 2019
a(n) = [x^n] (1 + x)^n * Product_{k>=2} 1 / (1 - x^k). - Ilya Gutkovskiy, Apr 24 2021

A292613 a(n) = [x^n] 1/(1-x)^n * Product_{k=1..n} 1/(1-x^k).

Original entry on oeis.org

1, 2, 7, 25, 92, 343, 1292, 4902, 18703, 71677, 275694, 1063636, 4114131, 15948762, 61946290, 241013869, 939125870, 3664299332, 14314777054, 55982787136, 219158088711, 858728875776, 3367576480747, 13216392846128, 51905939548950, 203989227456894, 802164259099114
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 20 2017

Keywords

Comments

Number of ways to pick n units in all partitions of 2n - Olivier Gérard, May 07 2020

Examples

			Illustration of comment for n=3, a(3)=25 :
Among the 11 integer partitions of 6, 3 have at least 3 ones.
3,1,1,1  ;  2,1,1,1,1;  1,1,1,1,1,1;
There are respectively 1, 4 and 20 ways to pick 3 of these.

Links

Crossrefs

Cf. A000041, A001700, A014329, A292463.
Cf. A292508, A322211.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1-x)^n*Product[1/(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]

Formula

a(n) ~ c * 4^n / sqrt(Pi*n), where c = 1/(2*QPochhammer[1/2, 1/2]) = 1.7313733097275318057689... - Vaclav Kotesovec, Sep 20 2017
a(n) = A292508(n,n+1). - Alois P. Heinz, Jul 16 2021

A292541 a(n) is n-th term of the Euler transform of -n,1,1,1,... .

Original entry on oeis.org

1, -1, 2, -3, 5, -9, 18, -39, 88, -200, 449, -988, 2131, -4527, 9540, -20090, 42510, -90596, 194299, -418105, 899493, -1929000, 4116944, -8742002, 18484225, -38974978, 82086786, -172927251, 364700265, -770223900, 1628602725, -3445907334, 7291399538
Offset: 0

Views

Author

Alois P. Heinz, Sep 18 2017

Keywords

Links

Crossrefs

Cf. A292463.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      binomial(k+n-1, n), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> b(n$2, -n):
seq(a(n), n=0..35);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)+k-1)*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n, -n):
seq(a(n), n=0..35);
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=1,
      combinat[numbpart](n), b(n, k+1)-b(n-1, k+1)))
    end:
a:= n-> b(n, -n):
seq(a(n), n=0..35);
  • Mathematica
    Table[SeriesCoefficient[(1 - x)^n*Product[1/(1 - x^k), {k, 2, n}], {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, May 07 2018 *)
b[n_, k_] := b[n, k] = If[n == 0, 1, If[k == 1, PartitionsP[n], b[n, k + 1] - b[n - 1, k + 1]]]; Table[b[n, -n], {n, 0, 40}] (* Vaclav Kotesovec, May 07 2018, after Alois P. Heinz *)

Formula

a(n) = [x^n] (1-x)^n / Product_{j=2..n} (1-x^j).
a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)/2 + Pi^2/96) * 2^(n - 1/2) / (sqrt(3)*n). - Vaclav Kotesovec, May 07 2018

A304781 a(n) = [x^n] (1/(1 - x)^n)*Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 2, 6, 21, 75, 274, 1016, 3807, 14377, 54627, 208584, 799669, 3076167, 11867511, 45897145, 177888715, 690770763, 2686879415, 10466761637, 40828165464, 159453481037, 623427464093, 2439907421914, 9557831470082, 37472409664888, 147028505564603, 577302980976146
Offset: 0

Views

Author

Ilya Gutkovskiy, May 18 2018

Keywords

Comments

Number of partitions of n into odd parts with n + 1 kinds of 1.

Links

Crossrefs

Cf. A000009, A036469, A095944, A128566, A128593, A292463, A292613, A293467.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - x)^n Product[(1 + x^k), {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x)^n Product[1/(1 - x^(2 k - 1)), {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x)^n Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[QPochhammer[-1, x]/(2 (1 - x)^n), {x, 0, n}], {n, 0, 26}]

Formula

a(n) = [x^n] (1/(1 - x)^n)*Product_{k>=1} 1/(1 - x^(2*k-1)).
a(n) = [x^n] (1/(1 - x)^n)*exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))).
a(n) ~ QPochhammer[-1, 1/2] * 4^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, May 18 2018
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.