A292835 Triangle C(nu,m) read by rows: the number of n-color odd compositions of nu into m parts.

1, 0, 1, 0, 0, 1, 0, 3, 0, 1, 0, 0, 6, 0, 1, 0, 5, 0, 9, 0, 1, 0, 0, 19, 0, 12, 0, 1, 0, 7, 0, 42, 0, 15, 0, 1, 0, 0, 44, 0, 74, 0, 18, 0, 1, 0, 9, 0, 138, 0, 115, 0, 21, 0, 1, 0, 0, 85, 0, 316, 0, 165, 0, 24, 0, 1, 0, 11, 0, 363, 0, 605, 0, 224, 0, 27, 0, 1, 0, 0, 146, 0, 1059, 0, 1032, 0, 292, 0, 30, 0, 1

Offset: 0

Author

R. J. Mathar, Sep 24 2017

Keywords

• nonn
• tabl
• easy

Examples

			The triangle starts in row nu=0 with columns 0<=m<=nu as
1;
0,1;
0,0,1;
0,3,0,1;
0,0,6,0,1;
0,5,0,9,0,1;
0,0,19,0,12,0,1;
0,7,0,42,0,15,0,1;
0,0,44,0,74,0,18,0,1;
0,9,0,138,0,115,0,21,0,1;
0,0,85,0,316,0,165,0,24,0,1;
0,11,0,363,0,605,0,224,0,27,0,1;
		

Links

• Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (7).
• Y.-h. Guo, n-Color Odd Self-Inverse Compositions, J. Int. Seq. 17 (2014) # 14.10.5, eq (1).

Crossrefs

Cf. A119749 (row sums), A005900 (column m=3), A300437.

Programs

Maple

A292835 := proc(nu,m)
    local a,j,i ;
    a := 0 ;
    for j from 0 to m do
        i := (nu-m)/2-j ;
        if type(i,integer) then
            a := a+binomial(2*m+i-1,2*m-1)*binomial(m,j) ;
        end if;
    end do:
    a;
end proc:
seq(seq(A292835(nu,m),m=0..nu),nu=0..12) ;

Mathematica

A292835[nu_, m_] := Module[{a, j, i}, a = 0; For[j = 0, j <= m, j++, i = (nu - m)/2 - j; If[IntegerQ[i], a = a + Binomial[2m + i - 1, 2m - 1]* Binomial[m, j]]]; a];
Table[Table[A292835[nu, m], {m, 0, nu}], {nu, 0, 12}] // Flatten (* Jean-François Alcover, Aug 15 2023, after Maple code *)