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A262072 Number T(n,k) of partitions of an n-set with distinct block sizes and maximal block size equal to k; triangle T(n,k), n>=0, ceiling((sqrt(1+8*n)-1)/2)<=k<=n, read by rows. 5
1, 1, 1, 3, 1, 4, 1, 10, 5, 1, 60, 15, 6, 1, 140, 21, 7, 1, 280, 224, 28, 8, 1, 1260, 630, 336, 36, 9, 1, 12600, 3780, 1050, 480, 45, 10, 1, 34650, 7392, 1650, 660, 55, 11, 1, 110880, 74844, 12672, 2475, 880, 66, 12, 1, 360360, 276276, 140712, 20592, 3575, 1144, 78, 13, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

EXAMPLE

Triangle T(n,k) begins:

: 1;

:    1;

:       1;

:       3,  1;

:           4,     1;

:          10,     5,    1;

:          60,    15,    6,    1;

:                140,   21,    7,   1;

:                280,  224,   28,   8,  1;

:               1260,  630,  336,  36,  9,  1;

:              12600, 3780, 1050, 480, 45, 10, 1;

MAPLE

b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,

       b(n, i-1) +`if`(i>n, 0, binomial(n, i)*b(n-i, i-1))))

    end:

T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):

seq(seq(T(n, k), k=ceil((sqrt(1+8*n)-1)/2)..n), n=0..14);

MATHEMATICA

b[n_, i_] := b[n, i] = If[i*(i+1)/2<n, 0, If[n==0, 1, b[n, i-1] + If[i>n, 0, Binomial[n, i]*b[n-i, i-1]]]]; T[n_, k_] := b[n, k] - If[k == 0, 0, b[n, k-1]]; Table[T[n, k], {n, 0, 14}, {k, Ceiling[(Sqrt[1+8*n]-1)/2], n}] // Flatten (* Jean-Fran├žois Alcover, Feb 04 2017, translated from Maple *)

CROSSREFS

Row sums give A007837.

Column sums give A262073.

Cf. A002024, A262071, A262078 (same read by columns).

Sequence in context: A180062 A079546 A014413 * A321743 A131632 A051348

Adjacent sequences:  A262069 A262070 A262071 * A262073 A262074 A262075

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Sep 10 2015

STATUS

approved

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Last modified February 19 00:35 EST 2020. Contains 332028 sequences. (Running on oeis4.)