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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified December 13 22:55 EST 2019. Contains 329974 sequences. (Running on oeis4.)