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 1-10 of 14 results. Next

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

A292622 Number A(n,k) of partitions of n with up to k distinct 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, 1, 1, 1, 3, 2, 2, 2, 1, 4, 4, 3, 3, 2, 1, 5, 7, 5, 5, 4, 4, 1, 6, 11, 9, 8, 7, 6, 4, 1, 7, 16, 16, 13, 12, 10, 8, 7, 1, 8, 22, 27, 22, 20, 17, 14, 11, 8, 1, 9, 29, 43, 38, 33, 29, 24, 19, 15, 12, 1, 10, 37, 65, 65, 55, 49, 41, 33, 26, 20, 14
Offset: 0

Views

Author

Alois P. Heinz, Sep 20 2017

Keywords

Comments

For fixed k>=0, A(n,k) ~ Pi * 2^(k - 5/2) * exp(Pi*sqrt(2*n/3)) / (3 * n^(3/2)). - Vaclav Kotesovec, Oct 24 2018

Examples

			A(3,4) =  9: 3, 21a, 21b, 21c, 21d, 1a1b1c, 1a1b1d, 1a1c1d, 1b1c1d.
A(4,3) =  8: 4, 31a, 31b, 31c, 22, 21a1b, 21a1c, 21b1c.
A(4,4) = 13: 4, 31a, 31b, 31c, 31d, 22, 21a1b, 21a1c, 21a1d, 21b1c, 21b1d, 21c1d, 1a1b1c1d.
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,  1,  2,  4,  7, 11,  16,  22,  29, ...
  1,  2,  3,  5,  9, 16,  27,  43,  65, ...
  2,  3,  5,  8, 13, 22,  38,  65, 108, ...
  2,  4,  7, 12, 20, 33,  55,  93, 158, ...
  4,  6, 10, 17, 29, 49,  82, 137, 230, ...
  4,  8, 14, 24, 41, 70, 119, 201, 338, ...
  7, 11, 19, 33, 57, 98, 168, 287, 488, ...

Links

Crossrefs

Columns k=0-10 give: A002865, A027336, A320689, A320690, A320691, A320692, A320693, A320694, A320695, A320696, A320697.
Rows n=0-4 give: A000012, A001477, A000124(k-1) for k>0, A011826 for k>0.
Main diagonal gives A292507.
Cf. A292508, A292741, A292745.

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, 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 == 1, Binomial[k, n], If[i > n, 0, b[n - i, i, k]] + b[n, i - 1, k]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)

Formula

G.f. of column k: (1 + x)^k * Product_{j>=2} 1 / (1 - x^j). - Ilya Gutkovskiy, Apr 24 2021

A292746 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of partitions of n with exactly k kinds of 1's which are introduced in ascending order.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 3, 1, 2, 3, 8, 6, 1, 2, 5, 19, 26, 10, 1, 4, 7, 43, 97, 66, 15, 1, 4, 11, 93, 334, 361, 141, 21, 1, 7, 15, 197, 1095, 1778, 1066, 267, 28, 1, 8, 22, 409, 3482, 8207, 7108, 2668, 463, 36, 1, 12, 30, 840, 10855, 36310, 43747, 23116, 5909, 751, 45, 1
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2017

Keywords

Examples

			T(3,0) = 1: 3.
T(3,1) = 2: 21a, 1a1a1a.
T(3,2) = 3: 1a1a1b, 1a1b1a, 1a1b1b. (The two kinds of 1's are labeled 1a and 1b)
T(3,3) = 1: 1a1b1c.
Triangle T(n,k) begins:
  1;
  0,  1;
  1,  1,   1;
  1,  2,   3,    1;
  2,  3,   8,    6,    1;
  2,  5,  19,   26,   10,    1;
  4,  7,  43,   97,   66,   15,    1;
  4, 11,  93,  334,  361,  141,   21,   1;
  7, 15, 197, 1095, 1778, 1066,  267,  28,  1;
  8, 22, 409, 3482, 8207, 7108, 2668, 463, 36, 1;
  ...

Links

Crossrefs

Columns k=0-10 give: A002865, A000041(n-1) for n>0, A259401(n-2) for n>1, A320816, A320817, A320818, A320819, A320820, A320821, A320822, A320823.
Main diagonal and first lower diagonal give: A000012, A000217.
Row sums give A292503.
T(2n,n) gives A292747.
Cf. A292741, A292745.

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:
T:= (n, k)-> b(n$2, k)-`if`(k=0, 0, b(n$2, k-1)):
seq(seq(T(n, k), k=0..n), n=0..14);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      k^n, b(n, i-1, k) +b(n-i, min(i, n-i), k))
    end:
T:= (n, k)-> add((-1)^i*b(n$2, k-i)/((k-i)!*i!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..14);
  • 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}]];
T[n_, k_] := b[n, n, k] - If[k == 0, 0, b[n, n, k - 1]];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 17 2018, translated from Maple *)

Formula

T(n,k) = A292745(n,k) - A292745(n,k-1) for k>0. T(n,0) = A292745(n,0) = A002865(n).
T(n,k) = Sum_{i=0..k} (-1)^i * A292741(n, k-i) / ((k-i)!*i!).

Extensions

Definition clarified by N. J. A. Sloane, Dec 12 2020

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 *)

A292741 Number A(n,k) of partitions of n with k sorts of part 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, 5, 3, 2, 1, 4, 10, 11, 5, 2, 1, 5, 17, 31, 24, 7, 4, 1, 6, 26, 69, 95, 50, 11, 4, 1, 7, 37, 131, 278, 287, 104, 15, 7, 1, 8, 50, 223, 657, 1114, 865, 212, 22, 8, 1, 9, 65, 351, 1340, 3287, 4460, 2599, 431, 30, 12, 1, 10, 82, 521, 2459, 8042, 16439, 17844, 7804, 870, 42, 14
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2017

Keywords

Examples

			A(1,3) = 3: 1a, 1b, 1c.
A(2,3) = 10: 2, 1a1a, 1a1b, 1a1c, 1b1a, 1b1b, 1b1c, 1c1a, 1c1b, 1c1c.
A(3,2) = 11: 3, 21a, 21b, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,     1,      1,      1, ...
  0,  1,   2,    3,     4,     5,      6,      7, ...
  1,  2,   5,   10,    17,    26,     37,     50, ...
  1,  3,  11,   31,    69,   131,    223,    351, ...
  2,  5,  24,   95,   278,   657,   1340,   2459, ...
  2,  7,  50,  287,  1114,  3287,   8042,  17215, ...
  4, 11, 104,  865,  4460, 16439,  48256, 120509, ...
  4, 15, 212, 2599, 17844, 82199, 289540, 843567, ...

Links

Crossrefs

Columns k=0-2 give: A002865, A000041, A090764.
Rows n=0-2 give: A000012, A001477, A002522, A071568.
Main diagonal gives A292462.
Cf. A003992, A004248, A009998, A051129, A292508, A292622, A292745.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0 or i<2, k^n,
      add(b(n-i*j, i-1, k), j=0..iquo(n, i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);
  • Mathematica
    b[0, , ] = 1; b[n_, i_, k_] := b[n, i, k] = If[i < 2, k^n, Sum[b[n - i*j, i - 1, k], {j, 0, Quotient[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 19 2018, translated from Maple *)

Formula

G.f. of column k: 1/(1-k*x) * 1/Product_{j>=2} (1-x^j).
A(n,k) = Sum_{j=0..n} A002865(j) * k^(n-j).

A320734 Number of partitions of n with three sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 1, 3, 7, 19, 52, 151, 442, 1314, 3921, 11737, 35171, 105464, 316318, 948863, 2846461, 8539221, 25617443, 76852054, 230555794, 691666924, 2075000173, 6224999772, 18674998357, 56024993883, 168074980137, 504224938548, 1512674813304, 4538024437036
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2018

Keywords

Links

Crossrefs

Column k=3 of A292745.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..3), add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, 3}], Sum[b[n - i j, i - 1], {j, 0, n/i}]];
a[n_] := b[n, n];
a /@ Range[0, 40] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A320735 Number of partitions of n with four sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 1, 3, 7, 20, 62, 217, 803, 3092, 12128, 48047, 191266, 763249, 3049383, 12190360, 48747140, 194960047, 779783252, 3119019290, 12475849884, 49902945245, 199610872683, 798441674561, 3193763066392, 12775045002551, 51100165484967, 204400632890492
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2018

Keywords

Links

Crossrefs

Column k=4 of A292745.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..4), add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, 4}], Sum[b[n - i j, i - 1], {j, 0, n/i}]];
a[n_] := b[n, n];
a /@ Range[0, 40] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A320736 Number of partitions of n with five sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 1, 3, 7, 20, 63, 232, 944, 4158, 19236, 91794, 446311, 2194569, 10863768, 53995350, 269013587, 1342192961, 6702368648, 33486112079, 167353481065, 836536395240, 4181989400979, 20907870188551, 104533122311131, 522646929294281, 2613178606952285
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2018

Keywords

Links

Crossrefs

Column k=5 of A292745.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..5), add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, 5}], Sum[b[n - i j, i - 1], {j, 0, n/i}]];
a[n_] := b[n, n];
a /@ Range[0, 40] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A320737 Number of partitions of n with six sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 1, 3, 7, 20, 63, 233, 965, 4425, 21904, 114910, 628754, 3544272, 20393306, 118986963, 700768255, 4152987416, 24714368292, 147480695339, 881688073414, 5277421580515, 31613933962624, 189481916086717, 1136086826214117, 6813308511956936, 40867019987219945
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2018

Keywords

Links

Crossrefs

Column k=6 of A292745.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..6), add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, 6}], Sum[b[n - i j, i - 1], {j, 0, n/i}]];
a[n_] := b[n, n];
a /@ Range[0, 40] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A320738 Number of partitions of n with seven sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 1, 3, 7, 20, 63, 233, 966, 4453, 22367, 120819, 693233, 4178068, 26179581, 169020426, 1115994109, 7491323062, 50893512269, 348746702822, 2404544709055, 16651752622351, 115675136440751, 805342277995251, 5615683405472021, 39202038270665250, 273878789880840798
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2018

Keywords

Links

Crossrefs

Column k=7 of A292745.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..7), add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, 7}], Sum[b[n - i j, i - 1], {j, 0, n/i}]];
a[n_] := b[n, n];
a /@ Range[0, 40] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)
Showing 1-10 of 14 results. Next

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