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A105477 Triangle read by rows: T(n,k) is the number of compositions of n into k parts when there are two kinds of part 2. 2
1, 2, 1, 1, 4, 1, 1, 6, 6, 1, 1, 6, 15, 8, 1, 1, 7, 23, 28, 10, 1, 1, 8, 30, 60, 45, 12, 1, 1, 9, 39, 98, 125, 66, 14, 1, 1, 10, 49, 144, 255, 226, 91, 16, 1, 1, 11, 60, 202, 437, 561, 371, 120, 18, 1, 1, 12, 72, 272, 685, 1128, 1092, 568, 153, 20, 1, 1, 13, 85, 355, 1015, 1995, 2555 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Triangle T(n,k), 1<=k<=n, given by (0, 2, -3/2, -1/6, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. Triangle T(n,k), 0<=k<=n, is the Riordan array (1, x*(1+x-x^2)/(1-x)) . - Philippe Deléham, Jan 25 2012

LINKS

Table of n, a(n) for n=1..73.

FORMULA

G.f.=tz(1+z-z^2)/(1-z-tz-tz^2+tz^3).

T(n,k)=Sum(binomial(k,j)*binomial(n-2j-1, k-j-1), j=0..n-k). - Emeric Deutsch, Aug 06 2006

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) - T(n-3,k-1), n>1. - Philippe Deléham, Jan 25 2012

EXAMPLE

T(4,2)=6 because we have (1,3),(3,1),(2,2),(2,2'),(2',2) and (2',2').

Triangle begins:

1;

2,1;

1,4,1;

1,6,6,1;

1,6,15,8,1;

Triangle T(n,k) given by (0,2,-3/2,-1/6,2/3,0,0,0,...) DELTA (1,0,0,0,0,...) begins :

1

0, 1

0, 2, 1

0, 1, 4, 1

0, 1, 6, 6, 1

0, 1, 6, 15, 8, 1...

MAPLE

G:=t*z*(1+z-z^2)/(1-z-t*z-t*z^2+t*z^3): Gser:=simplify(series(G, z=0, 15)): for n from 1 to 14 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 13 do seq(coeff(P[n], t^k), k=1..n) od; # yields sequence in triangular form

CROSSREFS

Row sums yield A077998.

Diagonals : A000012, A005843, A000384

Sequence in context: A122578 A208648 A005131 * A325772 A226174 A208482

Adjacent sequences:  A105474 A105475 A105476 * A105478 A105479 A105480

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Apr 09 2005

STATUS

approved

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Last modified December 7 09:33 EST 2019. Contains 329843 sequences. (Running on oeis4.)