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
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.
Programs
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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) ;
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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.
Comments