A272514 -id:A272514 - cp's OEIS frontend

A131632 Triangle T(n,k) read by rows = number of partitions of n-set into k blocks with distinct sizes, k = 1..A003056(n).

1, 1, 1, 3, 1, 4, 1, 15, 1, 21, 60, 1, 63, 105, 1, 92, 448, 1, 255, 2016, 1, 385, 4980, 12600, 1, 1023, 15675, 27720, 1, 1585, 61644, 138600, 1, 4095, 155155, 643500, 1, 6475, 482573, 4408404, 1, 16383, 1733550, 12687675, 37837800, 1, 26332, 4549808, 60780720

Offset: 1

Vladeta Jovovic, Sep 04 2007

Row sums = A007837.
Sum k! * T(n,k) = A032011.
Sum k * T(n,k) = A131623. - Geoffrey Critzer, Aug 30 2012.
T(n,k) is also the number of words w of length n over a k-ary alphabet {a1,a2,...,ak} with #(w,a1) > #(w,a2) > ... > #(w,ak) > 0, where #(w,x) counts the letters x in word w.  T(5,2) = 15: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, abbaa, baaaa, baaab, baaba, babaa, bbaaa. - Alois P. Heinz, Jun 21 2013

			Triangle T(n,k)begins:
  1;
  1;
  1,     3;
  1,     4;
  1,    15;
  1,    21,      60;
  1,    63,     105;
  1,    92,     448;
  1,   255,    2016;
  1,   385,    4980,    12600;
  1,  1023,   15675,    27720;
  1,  1585,   61644,   138600;
  1,  4095,  155155,   643500;
  1,  6475,  482573,  4408404;
  1, 16383, 1733550, 12687675, 37837800;
  ...

Alois P. Heinz, Rows n = 1..500, flattened

Columns k=1-10 give: A000012, A272514, A272515, A272516, A272517, A272518, A272519, A272520, A272521, A272522.

Cf. A000217, A003056, A007837, A022915, A032011, A131623, A226873, A226874, A327803.

b:= proc(n, i, t, v) option remember; `if`(t=1, 1/(n+v)!,
      add(b(n-j, j, t-1, v+1)/(j+v)!, j=i..n/t))
    end:
T:= (n, k)->`if`(k*(k+1)/2>n, 0, n!*b(n-k*(k+1)/2, 0, k, 1)):
seq(seq(T(n, k), k=1..floor(sqrt(2+2*n)-1/2)), n=1..20);
# Alois P. Heinz, Jun 21 2013
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..20);  # Alois P. Heinz, Sep 27 2019

nn=10;p=Product[1+y x^i/i!,{i,1,nn}];Range[0,nn]! CoefficientList[ Series[p,{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, Aug 30 2012 *)

E.g.f.: Product_{n>=1} (1+y*x^n/n!).

T(A000217(n),n) = A022915(n). - Alois P. Heinz, Jul 03 2018

A028331 Elements to the right of the central elements of the even-Pascal triangle A028326 that are not 2.

Original entry on oeis.org

6, 8, 20, 10, 30, 12, 70, 42, 14, 112, 56, 16, 252, 168, 72, 18, 420, 240, 90, 20, 924, 660, 330, 110, 22, 1584, 990, 440, 132, 24, 3432, 2574, 1430, 572, 156, 26, 6006, 4004, 2002, 728, 182, 28, 12870, 10010, 6006, 2730, 910, 210, 30, 22880, 16016

Offset: 0

Mohammad K. Azarian

			This sequence represents the following portion of A028330(n,k), with x being the elements of A028329(n):
  x;
  .,  .;
  .,  x,  .;
  .,  .,  6,  .;
  .,  .,  x,  8,  .;
  .,  .,  ., 20, 10,   .;
  .,  .,  .,  x, 30,  12,   .;
  .,  .,  .,  ., 70,  42,  14,    .;
  .,  .,  .,  .,  x, 112,  56,   16,   .;
  .,  .,  .,  .,  ., 252, 168,   72,  18,   .;
  .,  .,  .,  .,  .,   x, 420,  240,  90,  20,   .;
  .,  .,  .,  .,  .,   ., 924,  660, 330, 110,  22,  .;
  .,  .,  .,  .,  .,   .,   x, 1584, 990, 440, 132, 24, .;
As an irregular triangle:
    6;
    8;
   20,  10;
   30,  12;
   70,  42,  14;
  112,  56,  16;
  252, 168,  72,  18;
  420, 240,  90,  20;
  924, 660, 330, 110, 22;

G. C. Greubel, Rows n = 0..100 of the irregular triangle, flattened

Cf. A028326, A028327, A028328, A028329, A028330, A028332.

Cf. A272514, A286033.

[2*Binomial(n+3,k): k in [Floor((n+5)/2)..n+2], n in [0..12]]; // G. C. Greubel, Jul 14 2024

Table[2*Binomial[n+3, k+2 +Floor[(n+1)/2]], {n,0,12}, {k,0,Floor[n/2] }]//Flatten (* G. C. Greubel, Jul 14 2024 *)

def A028326(n,k): return 2*binomial(n, k)
flatten([[A028326(n+1,k) for k in range(((n+3)//2), n+1)] for n in range(21)]) # G. C. Greubel, Jul 14 2024

From G. C. Greubel, Jul 14 2024: (Start)
T(n, k) = 2*binomial(n+3, k+2 + floor((n+1)/2)).
Sum_{k=0..floor(n/2)} T(n, k) = A272514(n+3).
Sum_{k=0..n} (-1)^k*T(2*n, k) = 2*A286033(n+2).
Sum_{k=0..n} (-1)^k*T(2*n+1, k) = binomial(2*n+4, n+2) + 2*(-1)^n.
(End)

More terms from James Sellers

cp's OEIS Frontend

A131632 Triangle T(n,k) read by rows = number of partitions of n-set into k blocks with distinct sizes, k = 1..A003056(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula