OFFSET
0,2
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.
LINKS
Y.-h. Guo, n-Color Odd Self-Inverse Compositions, J. Int. Seq. 17 (2014) # 14.10.5, Table 1.
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.
EXAMPLE
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;
MAPLE
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 *)
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, Mar 05 2018
STATUS
approved