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 1-6 of 6 results.

A292745 Number A(n,k) of partitions of n with k sorts of part 1 which are introduced in ascending order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 2, 1, 1, 3, 6, 5, 2, 1, 1, 3, 7, 13, 7, 4, 1, 1, 3, 7, 19, 26, 11, 4, 1, 1, 3, 7, 20, 52, 54, 15, 7, 1, 1, 3, 7, 20, 62, 151, 108, 22, 8, 1, 1, 3, 7, 20, 63, 217, 442, 219, 30, 12, 1, 1, 3, 7, 20, 63, 232, 803, 1314, 439, 42, 14
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2017

Keywords

Examples

			A(3,2) = 6: 3, 21a, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,    1,    1,    1,    1,    1, ...
  0,  1,   1,    1,    1,    1,    1,    1,    1, ...
  1,  2,   3,    3,    3,    3,    3,    3,    3, ...
  1,  3,   6,    7,    7,    7,    7,    7,    7, ...
  2,  5,  13,   19,   20,   20,   20,   20,   20, ...
  2,  7,  26,   52,   62,   63,   63,   63,   63, ...
  4, 11,  54,  151,  217,  232,  233,  233,  233, ...
  4, 15, 108,  442,  803,  944,  965,  966,  966, ...
  7, 22, 219, 1314, 3092, 4158, 4425, 4453, 4454, ...

Links

Crossrefs

Columns k=0-10 give: A002865, A000041, A320733, A320734, A320735, A320736, A320737, A320738, A320739, A320740, A320741.
Main diagonal gives A292503.
Cf. A292508, A292622, A292741, 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, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=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}]];
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 Maple *)

Formula

A(n,k) = Sum_{j=0..k} A292746(n,j).
A(n,k) = A(n,n) for all k >= n.

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

A292463 Number of partitions of n with n kinds of 1.

Original entry on oeis.org

1, 1, 4, 14, 51, 188, 702, 2644, 10026, 38223, 146359, 562456, 2168134, 8379539, 32459199, 125984039, 489837300, 1907490728, 7438346255, 29042470132, 113522618066, 444199913556, 1739735079466, 6819657196928, 26753893533257, 105034060120469, 412637434996367
Offset: 0

Views

Author

Alois P. Heinz, Sep 16 2017

Keywords

Examples

			a(2) = 4: 2, 1a1a, 1a1b, 1b1b.

Links

Crossrefs

Main diagonal of A292508.
Cf. A000041, A014329, A292462, A292503, A292507, A292541.

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$3):
seq(a(n), n=0..30);
# 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$2):
seq(a(n), n=0..30);
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=1,
      combinat[numbpart](n), b(n-1, k) +b(n, k-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
  • Mathematica
    Table[SeriesCoefficient[1/(1-x)^(n-1) * Product[1/(1-x^k), {k,1,n}], {x,0,n}], {n,0,30}] (* Vaclav Kotesovec, Sep 19 2017 *)

Formula

a(n) = [x^n] 1/(1-x)^n * 1/Product_{j=2..n} (1-x^j).
a(n) is n-th term of the Euler transform of n,1,1,1,... .
a(n) ~ c * 4^n / sqrt(n), where c = QPochhammer[-1, 1/2] / (8*sqrt(Pi) * QPochhammer[1/4, 1/4]) = 0.48841139329043831428669851139824427133317... - Vaclav Kotesovec, Sep 19 2017
Equivalently, c = 1/(4*sqrt(Pi)*QPochhammer(1/2)). - Vaclav Kotesovec, Mar 17 2024

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

A014326 Convolution of partition numbers and Bell numbers.

Original entry on oeis.org

1, 2, 5, 12, 32, 95, 328, 1294, 5748, 28152, 149768, 856130, 5218107, 33712600, 229800588, 1646316230, 12355374717, 96861192976, 791258805462, 6720627186126, 59234364203973, 540812222400025, 5106663817693176, 49798678281859244, 500857393911224861
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000041, A000110.
Partial sums of A292503.

Programs

  • Magma
    A014326:= func< n | (&+[NumberOfPartitions(j)*Bell(n-j): j in [0..n]]) >;
[A014326(n): n in [0..40]]; // G. C. Greubel, Jan 08 2023
  • Maple
    with(combinat):
a:= n-> add(numbpart(k)*bell(n-k), k=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 15 2015
  • Mathematica
    a[n_]:= Sum[PartitionsP[k]*BellB[n-k], {k,0,n}];
Table[a[n], {n,0,30}] (* Jean-François Alcover, Dec 06 2016 *)
  • SageMath
    def A014326(n): return sum(number_of_partitions(j)*bell_number(n-j) for j in range(n+1))
[A014326(n) for n in range(41)] # G. C. Greubel, Jan 08 2023
Showing 1-6 of 6 results.

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

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