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 1-5 of 5 results.

A319298 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by increasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 3, 1, 7, 7, 1, 21, 25, 13, 1, 66, 101, 71, 21, 1, 258, 366, 396, 166, 31, 1, 1079, 1555, 1877, 1247, 337, 43, 1, 4987, 7099, 9199, 7855, 3305, 617, 57, 1, 25195, 34627, 47371, 47245, 27085, 7681, 1045, 73, 1, 136723, 184033, 253108, 284968, 203278, 79756, 16126, 1666, 91, 1
Offset: 1

Views

Author

Alois P. Heinz, Dec 07 2018

Keywords

Examples

			The 5 set partitions of {1,2,3} are:
  1   |2  |3
  1   |23
  2   |13
  3   |12
  123
so there are 7 elements in the first (smallest) blocks, 7 in the second blocks and only 1 in the third blocks.
Triangle T(n,k) begins:
      1;
      3,     1;
      7,     7,     1;
     21,    25,    13,     1;
     66,   101,    71,    21,     1;
    258,   366,   396,   166,    31,    1;
   1079,  1555,  1877,  1247,   337,   43,    1;
   4987,  7099,  9199,  7855,  3305,  617,   57,  1;
  25195, 34627, 47371, 47245, 27085, 7681, 1045, 73, 1;
  ...

Links

Crossrefs

Column k=1-10 gives A097147, A332942, A332943, A332944, A332945, A332946, A332947, A332948, A332949, A332950.
Row sums give A070071.
Cf. A319375, A322383.

Programs

  • Maple
    b:= proc(n, l) option remember; `if`(n=0, add(l[i]*
      x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
      b(n-j, sort([l[], j])), j=1..n))
    end:
T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
seq(T(n), n=1..12);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(b(n-i*j, i+1,
      max(0, t-j))/j!*combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
    end:
T:= (n, k)-> b(n, 1, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 02 2020
  • Mathematica
    b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[i]] x^i, {i, 1, Length[l]}], Sum[ Binomial[n-1, j-1] b[n-j, Sort[Append[l, j]]], {j, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, {}]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 28 2018, after Alois P. Heinz *)

A097148 Total sum of maximum block sizes in all partitions of n-set.

Original entry on oeis.org

1, 3, 10, 35, 136, 577, 2682, 13435, 72310, 414761, 2524666, 16239115, 109976478, 781672543, 5814797281, 45155050875, 365223239372, 3070422740989, 26780417126048, 241927307839731, 2260138776632752, 21805163768404127, 216970086170175575, 2224040977932468379
Offset: 1

Views

Author

Vladeta Jovovic, Jul 27 2004

Keywords

Comments

Let M be the infinite lower triangular matrix given by A080510 and v the column vector [1,2,3,...] then M*v=A097148 (this sequence, as column vector). - Gary W. Adamson, Feb 24 2011

Links

Crossrefs

Cf. A028417, A028418, A046746, A006128, A097145-A097147.
Cf. A080510.
Column k=1 of A319375.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0, m, add(
      b(n-j, max(j, m))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..24);  # Alois P. Heinz, Mar 02 2020, revised May 08 2024
  • Mathematica
    Drop[ Range[0, 22]! CoefficientList[ Series[ Sum[E^(E^x - 1) - E^Sum[x^j/j!, {j, 1, k}], {k, 0, 22}], {x, 0, 22}], x], 1] (* Robert G. Wilson v, Aug 05 2004 *)

Formula

E.g.f.: Sum_{k>=0} (exp(exp(x)-1)-exp(Sum_{j=1..k} x^j/j!)).

Extensions

More terms from Robert G. Wilson v, Aug 05 2004

A097146 Total sum of maximum list sizes in all sets of lists of n-set, cf. A000262.

Original entry on oeis.org

0, 1, 5, 31, 217, 1781, 16501, 172915, 1998641, 25468777, 352751941, 5292123431, 85297925065, 1472161501981, 27039872306357, 527253067633531, 10865963240550241, 236088078855319505, 5390956470528548101, 129102989125943058607, 3234053809095307670201, 84596120521251178630981, 2305894874979300173268085
Offset: 0

Views

Author

Vladeta Jovovic, Jul 27 2004

Keywords

Examples

			For n=4 we have 73 sets of lists (cf. A000262): (1234) (24 ways), (123)(4) (6*4 ways), (12)(34) (3*4 ways), (12)(3)(4) (6*2 ways), (1)(2)(3)(4) (1 way); so a(4)= 24*4+24*3+12*2+12*2+1*1 = 217.

Links

Crossrefs

Cf. A028417, A028418, A046746, A006128, A097145, A097147, A097148.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0, m, add(j!*
      b(n-j, max(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016
  • Mathematica
    b[n_, m_] := b[n, m] = If[n == 0, m, Sum[j! b[n-j, Max[m, j]] Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := b[n, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 05 2020, after Alois P. Heinz *)
  • PARI
    N=50; x='x+O('x^N);
egf=exp(x/(1-x))*sum(k=1,N, (1-exp(x^k/(x-1))) );
Vec( serlaplace(egf) ) /* show terms */

Formula

E.g.f.: exp(x/(1-x))*Sum_{k>0} (1-exp(x^k/(x-1))).

Extensions

a(0)=0 prepended by Alois P. Heinz, May 10 2016

A372650 Total sum over all partitions of [n] of the number of minimal blocks.

Original entry on oeis.org

0, 1, 3, 7, 27, 86, 393, 1688, 8291, 44143, 248428, 1480073, 9440049, 63265606, 444309232, 3273807272, 25227429123, 202458174614, 1689474026499, 14636685675142, 131413462612012, 1220636654904548, 11712836883408675, 115956109213769404, 1182944504931376337
Offset: 0

Views

Author

Alois P. Heinz, May 08 2024

Keywords

Examples

			a(4) = 27 = 1+1+1+2+2+1+2+2+2+1+2+2+2+2+4: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.

Links

Crossrefs

Cf. A000110, A097147, A372649, A372762.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, t,
      add(binomial(n-1, j-1)*b(n-j, min(j, m),
     `if`(j b(n$2, 0):
seq(a(n), n=0..24);
  • Mathematica
    b[n_, m_, t_] := b[n, m, t] = If[n == 0, t,
   Sum[Binomial[n - 1, j - 1]*b[n - j, Min[j, m],
   If[j < m, 1, If[j == m, t + 1, t]]], {j, 1, n}]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 11 2024, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} k * A372762(n,k).

A182931 Generalized Bell numbers; square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 1.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 5, 1, 0, 1, 15, 1, 0, 0, 1, 52, 4, 1, 0, 0, 1, 203, 11, 1, 0, 0, 0, 1, 877, 41, 1, 1, 0, 0, 0, 1, 4140, 162, 11, 1, 0, 0, 0, 0, 1, 21147, 715, 36, 1, 1, 0, 0, 0, 0, 1, 115975, 3425, 92, 1, 1, 0, 0, 0, 0, 0, 1, 678570, 17722, 491, 36, 1, 1, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Peter Luschny, Apr 05 2011

Keywords

Examples

			Array starts:
[k=      1       2       3       4       5]
[n=0]    1,      1,      1,      1,      1,
[n=1]    1,      0,      0,      0,      0,
[n=2]    2,      1,      0,      0,      0,
[n=3]    5,      1,      1,      0,      0,
[n=4]   15,      4,      1,      1,      0,
[n=5]   52,     11,      1,      1,      1,
[n=6]  203,     41,     11,      1,      1,
[n=7]  877,    162,     36,      1,      1,
[n=8] 4140,    715,     92,     36,      1,
   A000110,A000296,A006505,A057837,A057814, ...

Links

Crossrefs

Cf. A000110, A000296, A006505, A057837, A057814, A097147.
Row sums are A097147 for n >= 1.

Programs

  • Maple
    egf := k -> exp(exp(x)*(1-GAMMA(k,x)/GAMMA(k)));
T := (n,k) -> n!*coeff(series(egf(k),x,n+1),x,n):
seq(print(seq(T(n,k),k=1..8)),n=0..8);
  • Mathematica
    egf[k_] := Exp[Exp[x] (1 - Gamma[k, x]/Gamma[k])];
T[n_, k_] := n! SeriesCoefficient[egf[k], {x, 0, n}];
Table[T[n-k+1, k], {n, 0, 11}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 13 2019 *)

Formula

E.g.f.: exp(exp(x)*(1-Gamma(k,x)/Gamma(k))); Gamma(k,x) the incomplete Gamma function.
Showing 1-5 of 5 results.

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