A261800 Number of 8-compositions of n: matrices with 8 rows of nonnegative integers with positive column sums and total element sum n.

1, 8, 100, 1208, 14554, 175352, 2112772, 25456328, 306717703, 3695574048, 44527157584, 536497912672, 6464145163032, 77885061063584, 938419943222768, 11306815168562400, 136233325153964242, 1641445323534504928, 19777413104380161776, 238293693669343744032

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 an octonary alphabet whose letters appear in alphabetical order.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..925
• Index entries for linear recurrences with constant coefficients, signature (16,-56,112,-140,112,-56,16,-2).

Crossrefs

Column k=8 of A261780.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*binomial(j+7, 7), j=1..n))
    end:
seq(a(n), n=0..20);

Mathematica

CoefficientList[Series[(1-x)^8/(2(1-x)^8-1),{x,0,30}],x] (* or  *) LinearRecurrence[{16,-56,112,-140,112,-56,16,-2},{1,8,100,1208,14554,175352,2112772,25456328,306717703},30] (* Harvey P. Dale, Jul 15 2023 *)

Formula

G.f.: (1-x)^8/(2*(1-x)^8-1).

a(n) = A261780(n,8).

a(n) = Sum_{k>=0} (1/2)^(k+1) * binomial(n-1+8*k,n). - Seiichi Manyama, Aug 06 2024