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Revision History for A243149

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Number of compositions of n such that the sum of the parts counted without multiplicities is equal to the sum of all multiplicities.
(history; published version)

#10 by Bruno Berselli at Mon May 21 08:04:53 EDT 2018

STATUS

reviewed

approved

#9 by Joerg Arndt at Mon May 21 06:47:59 EDT 2018

STATUS

proposed

reviewed

#8 by Jean-François Alcover at Mon May 21 05:49:47 EDT 2018

STATUS

editing

proposed

#7 by Jean-François Alcover at Mon May 21 05:49:43 EDT 2018

MATHEMATICA

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0, Expand[Sum[x^If[j == 0, 0, i - j]*b[n - i*j, i - 1, p + j]/j!, {j, 0, n/i}]]]];

a[n_] := Coefficient[b[n, n, 0], x, 0];

Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

STATUS

approved

editing

#6 by Alois P. Heinz at Sun Jun 01 03:27:52 EDT 2014

STATUS

editing

approved

#5 by Alois P. Heinz at Sun Jun 01 03:26:13 EDT 2014

MAPLE

b:= proc(n, i, p) option remember; `if`(n=0, p!,

`if`(i<1, 0, expand(add(x^`if`(j=0, 0, i-j)*

b(n-i*j, i-1, p+j)/j!, j=0..n/i))))

end:

a:= n-> coeff(b(n$2, 0), x, 0):

seq(a(n), n=0..50);

#4 by Alois P. Heinz at Sun Jun 01 03:18:44 EDT 2014

LINKS

Alois P. Heinz, <a href="/A243149/b243149.txt">Table of n, a(n) for n = 0..200</a>

#3 by Alois P. Heinz at Fri May 30 22:32:13 EDT 2014

EXAMPLE

a(8) = 11: [1,1,3,3], [1,3,1,3], [1,3,3,1], [3,1,1,3], [3,1,3,1], [3,3,1,1], [1,1,1,1,4], [1,1,1,4,1], [1,1,4,1,1], [1,4,1,1,1], [4,1,1,1,1].

#2 by Alois P. Heinz at Fri May 30 22:16:22 EDT 2014

NAME

allocated for Alois P. Heinz

Number of compositions of n such that the sum of the parts counted without multiplicities is equal to the sum of all multiplicities.

DATA

1, 1, 0, 0, 4, 3, 4, 0, 11, 31, 70, 177, 242, 382, 482, 874, 1655, 4440, 10696, 24390, 49867, 95850, 172980, 289229, 492233, 811753, 1468084, 2813206, 5929361, 12780690, 27858421, 59275097, 122326098, 243179349, 467856049, 873044584, 1588187110, 2842593612

OFFSET

0,5

CROSSREFS

Cf. A114638 (the same for partitions).

KEYWORD

allocated

nonn

AUTHOR

Alois P. Heinz, May 30 2014

STATUS

approved

editing

#1 by Alois P. Heinz at Fri May 30 22:16:22 EDT 2014

NAME

allocated for Alois P. Heinz

KEYWORD

allocated

STATUS

approved