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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.

%I A300437 #10 Aug 15 2023 10:53:02
%S A300437 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,
%T A300437 15,72,110,96,43,20,3,1,17,104,206,200,143,52,23,3,1,19,145,346,442,
%U A300437 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
%N 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.
%C A300437 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.
%H A300437 Y.-h. Guo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Guo/guo9.html">n-Color Odd Self-Inverse Compositions</a>, J. Int. Seq. 17 (2014) # 14.10.5, Table 1.
%F A300437 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.
%e A300437 The triangle starts in row nu=0 with columns 0<=m<=nu as:
%e A300437 1;
%e A300437 3,1;
%e A300437 5,3,1;
%e A300437 7,8,3,1;
%e A300437 9,16,11,3,1;
%e A300437 11,29,25,14,3,1;
%e A300437 13,47,58,34,17,3,1;
%e A300437 15,72,110,96,43,20,3,1;
%e A300437 17,104,206,200,143,52,23,3,1;
%e A300437 19,145,346,442,317,199,61,26,3,1;
%e A300437 21,195,571,822,807,461,264,70,29,3,1;
%e A300437 23,256,881,1565,1613,1328,632,338,79,32,3,1;
%e A300437 25,328,1337,2671,3478,2800,2032,830,421,88,35,3,1;
%e A300437 27,413,1939,4596,6402,6742,4464,2946,1055,513,97,38,3,1;
%p A300437 A300437 := proc(k,l)
%p A300437     local a,t,i,j ;
%p A300437     a := 0 ;
%p A300437     for t from 1 to 2*k+1 by 2 do
%p A300437         for j from 0 to l do
%p A300437             i := (2*k+1-t-2*l)/4-j ;
%p A300437             if type(i,'integer') then
%p A300437                 a := a+t*binomial(2*l+i-1,2*l-1)*binomial(l,j) ;
%p A300437             end if;
%p A300437         end do:
%p A300437     end do:
%p A300437     a ;
%p A300437 end proc:
%p A300437 seq(seq(A300437(k,l),l=0..k),k=0..13) ;
%t A300437 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];
%t A300437 Table[Table[A300437[k, l], {l, 0, k}], {k, 0, 13}] // Flatten (* _Jean-François Alcover_, Aug 15 2023, after Maple code *)
%Y A300437 Cf. A131941 (column 2?), A300438 (row sums), A292835.
%K A300437 nonn,tabl,easy
%O A300437 0,2
%A A300437 _R. J. Mathar_, Mar 05 2018