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A278644 Number of partitions of n into parts of sorts {1, 2, ... }. 3
1, 1, 4, 17, 95, 649, 5423, 53345, 604570, 7744990, 110596370, 1740967790, 29943077149, 558541778035, 11229820022013, 242071441524480, 5568954194762675, 136181762611151941, 3527284819779421843, 96465042641948254298, 2777679881076121497601 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Parts are unordered, sorts are ordered, all sorts up to the highest have to be present.

a(n) mod 2 = A040051(n).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..424

FORMULA

a(n) = Sum_{k=0..n} A255970(n,k).

EXAMPLE

a(3) = 17: 1a1a1a, 2a1a, 1a, 1a1a1b, 1a1b1a, 1b1a1a, 1b1b1a, 1b1a1b, 1a1b1b, 2a1b, 2b1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a (in this example the sorts are labeled a, b, c).

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

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

    end:

a:= n-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):

seq(a(n), n=0..25);

MATHEMATICA

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; a[n_] := Sum[Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-Fran├žois Alcover, Feb 06 2017, translated from Maple *)

CROSSREFS

Row sums of A255970.

Cf. A040051, A258466.

Sequence in context: A323664 A067084 A123750 * A249078 A353546 A024052

Adjacent sequences:  A278641 A278642 A278643 * A278645 A278646 A278647

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Nov 24 2016

STATUS

approved

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Last modified August 15 11:04 EDT 2022. Contains 356145 sequences. (Running on oeis4.)