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.

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

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. = t*z*(1+z-z^2)/(1-z-t*z-t*z^2+t*z^3).

T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-2j-1, k-j-1). - 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;
From Philippe Deléham, Jan 25 2012: (Start)
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; ... (End)

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