A261802 Number of 10-compositions of n: matrices with 10 rows of nonnegative integers with positive column sums and total element sum n.

1, 10, 155, 2320, 34640, 517252, 7723970, 115339960, 1722340115, 25719233330, 384058268507, 5735036957760, 85639736481880, 1278834734405320, 19096488909285540, 285162639746429024, 4258255614078447290, 63587365059302801520, 949532710487622388080

Offset: 0

Author

Alois P. Heinz, Sep 01 2015

Keywords

• nonn
• easy

Comments

Also the number of compositions of n where each part i is marked with a word of length i over a denary alphabet whose letters appear in alphabetical order.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..850
• Index entries for linear recurrences with constant coefficients, signature (20, -90, 240, -420, 504, -420, 240, -90, 20, -2).

Crossrefs

Column k=10 of A261780.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*binomial(j+9, 9), j=1..n))
    end:
seq(a(n), n=0..20);

Formula

G.f.: (1-x)^10/(2*(1-x)^10-1).

a(n) = A261780(n,10).

a(n) = Sum_{k>=0} (1/2)^(k+1) * binomial(n-1+10*k,n). - Seiichi Manyama, Aug 06 2024