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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A229223 Number G(n,k) of set partitions of {1,...,n} into sets of size at most k; triangle G(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 10, 14, 15, 0, 1, 26, 46, 51, 52, 0, 1, 76, 166, 196, 202, 203, 0, 1, 232, 652, 827, 869, 876, 877, 0, 1, 764, 2780, 3795, 4075, 4131, 4139, 4140, 0, 1, 2620, 12644, 18755, 20645, 21065, 21137, 21146, 21147
Offset: 0

Views

Author

Alois P. Heinz, Sep 16 2013

Keywords

Comments

John Riordan calls these Allied Bell Numbers. - N. J. A. Sloane, Jan 10 2018
G(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. G(n,k) = G(n,n) = A000110(n) for k>n.
G(n,k) - G(n,k-1) = A080510(n,k).
A column G(n>=0,k) can be generated by a linear recurrence with polynomial coefficients, where the initial terms correspond with A000110, and the coefficients contain constant factors derived from A008279 (cf. recg(k) in the fourth Maple program below). - Georg Fischer, May 19 2021

Examples

			G(4,2) = 10: 1/2/3/4, 12/3/4, 13/2/4, 14/2/3, 1/23/4, 1/24/3, 1/2/34, 12/34, 13/24, 14/23.
Triangle G(n,k) begins:
  1;
  0,  1;
  0,  1,   2;
  0,  1,   4,    5;
  0,  1,  10,   14,   15,
  0,  1,  26,   46,   51,   52;
  0,  1,  76,  166,  196,  202,  203;
  0,  1, 232,  652,  827,  869,  876,  877;
  0,  1, 764, 2780, 3795, 4075, 4131, 4139, 4140;
  ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000012, A000085, A001680, A001681, A110038, A148092, A229224, A229225, A229226, A229227.
Main diagonal gives: A000110. Lower diagonal gives: A058692.
Cf. A066223 (G(2n,2)), A229228 (G(2n,n)), A229229 (G(n^2,n)), A227223 (G(2^n,n)).
Cf. A008279, A080510.

Programs

  • Maple
    G:= proc(n, k) option remember; `if`(n=0, 1, `if`(k<1, 0,
       add(G(n-k*j, k-1) *n!/k!^j/(n-k*j)!/j!, j=0..n/k)))
    end:
seq(seq(G(n, k), k=0..n), n=0..10);
# second Maple program:
G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j,k) od; % fi
    end:
seq(seq(G(n, k), k=0..n), n=0..10);
# third Maple program:
G:= proc(n, k) option remember; `if`(n=0, 1, add(
      G(n-i, k)*binomial(n-1, i-1), i=1..min(n, k)))
    end:
seq(seq(G(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 26 2017
# fourth Maple program (for columns G(n>=0,k)):
init := n -> seq(a(j) = combinat:-bell(j), j=0..n): # A000110
b := (n, k) -> mul((n - j)/(j + 1), j = 0..k-1):
recg := k -> {(k-1)!*(add(j*b(n, j)*a(n-j), j = 1..k) - n*a(n)), init(k-1)}:
column := proc(k, len) local f; f := gfun:-rectoproc(recg(k), a(n), remember):
map(f, [$0..len-1]) end:
seq(print(column(k, 12)), k=1..9); # Georg Fischer, May 19 2021
  • Mathematica
    g[n_, k_] := g[n, k] = If[n == 0, 1, If[k < 1, 0, Sum[g[n - k*j, k - 1] *n!/k!^j/(n - k*j)!/j!, { j, 0, n/k}]]]; Table[Table[g[n, k], { k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

Formula

G(0,k) = 1, G(n,k) = 0 for n>0 and k<1, otherwise G(n,k) = Sum_{j=0..floor(n/k)} G(n-k*j,k-1) * n!/(k!^j*(n-k*j)!*j!).
G(n,k) = G(n-1,k) +(n-1)/1 *(G(n-2,k) +(n-2)/2 *(G(n-3,k) +(n-3)/3 *(G(n-4,k) + ... +(n-(k-1))/(k-1) *G(n-k,k)...))).
E.g.f. of column k: exp(Sum_{j=1..k} x^j/j!).

A176230 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].

Original entry on oeis.org

1, 1, 1, 3, 6, 1, 15, 45, 15, 1, 105, 420, 210, 28, 1, 945, 4725, 3150, 630, 45, 1, 10395, 62370, 51975, 13860, 1485, 66, 1, 135135, 945945, 945945, 315315, 45045, 3003, 91, 1, 2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1, 34459425
Offset: 0

Views

Author

Paul Barry, Apr 12 2010

Keywords

Comments

Row sums are A066223. Reverse of A119743. Inverse is alternating sign version.
Diagonal sums are essentially A025164.
From Tom Copeland, Dec 13 2015: (Start)
See A099174 for relations to the Hermite polynomials and the link for operator relations, including the infinitesimal generator containing A000384.
Row polynomials are 2^n n! Lag(n,-x/2,-1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q.
The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.
Divided along the diagonals by the initial element (A001147) of the diagonal, this matrix becomes the even rows of A034839.
(End)
The first few rows appear in expansions related to the Dedekind eta function on pp. 537-538 of the Chan et al. link. - Tom Copeland, Dec 14 2016

Examples

			Triangle begins
        1,
        1,        1,
        3,        6,        1,
       15,       45,       15,       1,
      105,      420,      210,      28,       1,
      945,     4725,     3150,     630,      45,      1,
    10395,    62370,    51975,   13860,    1485,     66,    1,
   135135,   945945,   945945,  315315,   45045,   3003,   91,   1,
  2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1
Production matrix is
  1,  1,
  2,  5,  1,
  0, 12,  9,  1,
  0,  0, 30, 13,  1,
  0,  0,  0, 56, 17,   1,
  0,  0,  0,  0, 90,  21,   1,
  0,  0,  0,  0,  0, 132,  25,   1,
  0,  0,  0,  0,  0,   0, 182,  29,  1,
  0,  0,  0,  0,  0,   0,   0, 240, 33, 1.

Links

Crossrefs

Cf. A113278.
Cf. A000384, A001147, A034839, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299.

Programs

  • Maple
    ser := n -> series(KummerU(-n, 1/2, x), x, n+1):
seq(seq((-2)^(n-k)*coeff(ser(n), x, k), k=0..n), n=0..8); # Peter Luschny, Jan 18 2020
  • Mathematica
    t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); u[n_, k_] := t[2 n, k + n]; Table[ u[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jan 14 2011 *)

Formula

Number triangle T(n,k) = (2n)!/((2k)!(n-k)!2^(n-k)).
T(n,k) = A122848(2n,k+n). - R. J. Mathar, Jan 14 2011
[x^(1/2)(1+2D)]^2 p(n,x)= p(n+1,x) and [D/(1+2D)]p(n,x)= n p(n-1,x) for the row polynomials of T, with D=d/dx. - Tom Copeland, Dec 26 2012
E.g.f.: exp[t*x/(1-2x)]/(1-2x)^(1/2). - Tom Copeland , Dec 10 2013
The n-th row polynomial R(n,x) is given by the type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} (2*k+1)*(2*k+3)*...*(2*k+1+2*(n-1))*(x/2)^k/k!. Cf. A113278. - Peter Bala, Jun 23 2014
The raising operator in my 2012 formula expanded is R = [x^(1/2)(1+2D)]^2 = 1 + x + (2 + 4x) D + 4x D^2, which in matrix form acting on an o.g.f. (formal power series) is the transpose of the production array below. The linear term x is the diagonal of ones after transposition. The main diagonal comes from (1 + 4xD) x^n = (1 + 4n) x^n. The last diagonal comes from (2 D + 4 x D^2) x^n = (2 + 4 xD) D x^n = n * (2 + 4(n-1)) x^(n-1). - Tom Copeland, Dec 13 2015
T(n, k) = (-2)^(n-k)*[x^k] KummerU(-n, 1/2, x). - Peter Luschny, Jan 18 2020

A229243 Number A(n,k) of set partitions of {1,...,k*n} into sets of size at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 10, 5, 1, 1, 1, 76, 166, 15, 1, 1, 1, 764, 12644, 3795, 52, 1, 1, 1, 9496, 1680592, 3305017, 112124, 203, 1, 1, 1, 140152, 341185496, 6631556521, 1245131903, 4163743, 877, 1
Offset: 0

Views

Author

Alois P. Heinz, Sep 17 2013

Keywords

Examples

			Square array A(n,k) begins:
  1,  1,      1,          1,              1,                   1, ...
  1,  1,      1,          1,              1,                   1, ...
  1,  2,     10,         76,            764,                9496, ...
  1,  5,    166,      12644,        1680592,           341185496, ...
  1, 15,   3795,    3305017,     6631556521,      25120541332271, ...
  1, 52, 112124, 1245131903, 41916097982471, 3282701194678476257, ...

Links

Crossrefs

Columns k=0-3 give: A000012, A000110, A229228, A229413.
Rows n=0+1, 2-3 give: A000012, A066223, A229414.
Main diagonal gives: A229229.
Cf. A229223.

Programs

  • Maple
    G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
A:= (n, k)-> G(n*k, n):
seq(seq(A(n, d-n), n=0..d), d=0..10);
  • Mathematica
    G[n_, k_] := G[n, k] = Module[{j, g}, Which[k > n, G[n, n], n == 0, 1, k < 1, 0, True, g = G[n-k, k]; For[j = k-1, j >= 1, j--, g = g*(n-j)/j + G[n-j, k] ]; g ] ]; A[n_, k_] := G[n*k, n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 23 2013, translated from Maple *)

Formula

A(n,k) = (n*k)! * [x^(n*k)] exp(Sum_{j=1..n} x^j/j!).
A(n,k) = A229223(n*k,n).

A066224 Bisection of A000085.

Original entry on oeis.org

1, 4, 26, 232, 2620, 35696, 568504, 10349536, 211799312, 4809701440, 119952692896, 3257843882624, 95680443760576, 3020676745975552, 101990226254706560, 3666624057550245376, 139813029266338603264
Offset: 0

Views

Author

N. J. A. Sloane, Dec 19 2001

Keywords

Crossrefs

Cf. A066223.
Unsigned row sums of A130757.

Programs

  • Maple
    a := proc(n) option remember: if n = 0 then RETURN(1) fi: if n = 1 then RETURN(1) fi: a(n-1)+(n-1)*a(n-2): end: for i from 1 to 61 by 2 do printf(`%d,`,a(i)) od: # James Sellers, Feb 11 2002
  • Mathematica
    Table[n! 2^n LaguerreL[n,1/2,-1/2],{n,0,20}] (* Harvey P. Dale, Mar 11 2013 *)
Table[(-2)^n HypergeometricU[-n, 3/2, -(1/2)], {n, 0, 90}] (* Emanuele Munarini, Aug 31 2017 *)

Formula

a(n) = n!*2^n*LaguerreL(n, 1/2, -1/2). - Vladeta Jovovic, May 10 2003
a(n) = sum(n!*(2^(n-m))*binomial(n+1/2,n-m)/m!,m=0..n), n>=0.
a(n) ~ n^(n+1/2)*2^n*exp(-n+sqrt(2*n)-1/4) * (1 + 19/(24*sqrt(2*n))). - Vaclav Kotesovec, Jun 22 2013
a(n+2) - 4*(n+2)*a(n+1) + 2*(n+1)*(2*n+3)*a(n) = 0 - Emanuele Munarini, Aug 31 2017

Extensions

More terms from James Sellers, Feb 11 2002
Showing 1-4 of 4 results.

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

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