A300437 Triangle T(nu,m) read by rows: The number of N-color odd self-inverse compositions of (2*nu+1) into (2*m+1) parts.

1, 3, 1, 5, 3, 1, 7, 8, 3, 1, 9, 16, 11, 3, 1, 11, 29, 25, 14, 3, 1, 13, 47, 58, 34, 17, 3, 1, 15, 72, 110, 96, 43, 20, 3, 1, 17, 104, 206, 200, 143, 52, 23, 3, 1, 19, 145, 346, 442, 317, 199, 61, 26, 3, 1, 21, 195, 571, 822, 807, 461, 264, 70, 29, 3, 1, 23, 256, 881, 1565, 1613, 1328, 632, 338, 79, 32, 3, 1

Offset: 0

Author

R. J. Mathar, Mar 05 2018

Keywords

• nonn
• tabl
• easy

Comments

Table 1 of Guo contains several typos which are not compliant with the formula on page 4 for S_o(2k+1,2l+1). Also the formula has been modified to read S_o(2k+1,2l+1) = sum_{t=1..2k+1) sum_{i+j= (2k+1-t-2l)/4} t*binomial(2l+i-1,2l-1)*binomial(l,j). So the upper limit on t has been extended and a factor t has been inserted.

Examples

			The triangle starts in row nu=0 with columns 0<=m<=nu as:
1;
3,1;
5,3,1;
7,8,3,1;
9,16,11,3,1;
11,29,25,14,3,1;
13,47,58,34,17,3,1;
15,72,110,96,43,20,3,1;
17,104,206,200,143,52,23,3,1;
19,145,346,442,317,199,61,26,3,1;
21,195,571,822,807,461,264,70,29,3,1;
23,256,881,1565,1613,1328,632,338,79,32,3,1;
25,328,1337,2671,3478,2800,2032,830,421,88,35,3,1;
27,413,1939,4596,6402,6742,4464,2946,1055,513,97,38,3,1;
		

Links

• Y.-h. Guo, n-Color Odd Self-Inverse Compositions, J. Int. Seq. 17 (2014) # 14.10.5, Table 1.

Crossrefs

Cf. A131941 (column 2?), A300438 (row sums), A292835.

Programs

Maple

A300437 := proc(k,l)
    local a,t,i,j ;
    a := 0 ;
    for t from 1 to 2*k+1 by 2 do
        for j from 0 to l do
            i := (2*k+1-t-2*l)/4-j ;
            if type(i,'integer') then
                a := a+t*binomial(2*l+i-1,2*l-1)*binomial(l,j) ;
            end if;
        end do:
    end do:
    a ;
end proc:
seq(seq(A300437(k,l),l=0..k),k=0..13) ;

Mathematica

A300437[k_, l_] := Module[{a, t, i, j }, a = 0; For[t = 1, t <= 2k + 1, t += 2, For[j = 0, j <= l, j++, i = (2k + 1 - t - 2*l)/4 - j; If[ IntegerQ[i], a = a + t*Binomial[2l + i - 1, 2l - 1]*Binomial[l, j]]]]; a];
Table[Table[A300437[k, l], {l, 0, k}], {k, 0, 13}] // Flatten (* Jean-François Alcover, Aug 15 2023, after Maple code *)

Formula

64*T(nu+2,2) = 51 +1306/15*nu +13*(-1)^nu +56/3*nu^3 +170/3*nu^2 +4/15*nu^5 +10*(-1)^nu*nu +2*(-1)^nu*nu^2 +10/3*nu^4 with g.f. (1+x^2)^2/[(1+x)^3*(1-x)^6], column 2.