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

Author

Emeric Deutsch, Apr 09 2005

Keywords

• nonn
• tabl

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

Examples

			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)
		

Crossrefs

Row sums yield A077998.

Diagonals: A000012, A005843, A000384.

Programs

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

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