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

1,1

Links

Table of n, a(n) for n=1..61.

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).

Example

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;

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 *)

Crossrefs

Row sums yield A052536.

Sequence in context: A173897 A152593 A243548 * A114427 A129355 A080963

Adjacent sequences: A105475 A105476 A105477 * A105479 A105480 A105481

Keyword

nonn,tabl

Author

Emeric Deutsch, Apr 10 2005

Status

approved