A105478 Triangle read by rows: T(n,k) is the number of compositions of n into k parts when parts 1 and 2 are of two kinds.

2, 2, 4, 1, 8, 8, 1, 8, 24, 16, 1, 8, 36, 64, 32, 1, 9, 44, 128, 160, 64, 1, 10, 54, 192, 400, 384, 128, 1, 11, 66, 264, 720, 1152, 896, 256, 1, 12, 79, 352, 1120, 2432, 3136, 2048, 512, 1, 13, 93, 456, 1632, 4272, 7616, 8192, 4608, 1024, 1, 14, 108, 576, 2280, 6816

Offset: 1

Author

Emeric Deutsch, Apr 10 2005

Keywords

• nonn
• tabl

Examples

			T(4,2)=8 because we have (1,3),(1',3),(3,1),(3,1'),(2,2),(2,2'),(2',2) and (2',2').
Triangle begins:
2;
2,4;
1,8,8;
1,8,24,16;
1,8,36,64,32;
		

Crossrefs

Row sums yield A052536.

Programs

Maple

G:=t*z*(2-z^2)/(1-z-2*t*z+t*z^3): Gser:=simplify(series(G,z=0,14)): for n from 1 to 12 do P[n]:=expand(coeff(Gser,z^n)) od: for n from 1 to 12 do seq(coeff(P[n],t^k),k=1..n) od; # yields sequence in triangular form

Mathematica

t[1, 1] = t[2, 1] = 2; t[3, 2] = 8; t[, 1] = 1; t[n, n_] := 2^n; t[n_, k_] /; 1 <= k <= n := t[n, k] = t[n-1, k] + 2*t[n-1, k-1] - t[n-3, k-1]; t[n_, k_] = 0; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 12 2013 *)

Formula

G.f.=tz(2-z^2)/(1-z-2tz+tz^3). T(n, k)=T(n-1, k)+2T(n-1, k-1)-T(n-3, k-1).