A105474 Triangle read by rows: T(n,k) is number of compositions of n into k parts when each odd part can be of two kinds.

1, 0, 2, 0, 1, 4, 0, 2, 4, 8, 0, 1, 9, 12, 16, 0, 2, 8, 30, 32, 32, 0, 1, 14, 37, 88, 80, 64, 0, 2, 12, 66, 136, 240, 192, 128, 0, 1, 19, 75, 257, 440, 624, 448, 256, 0, 2, 16, 116, 352, 890, 1312, 1568, 1024, 512, 0, 1, 24, 126, 564, 1401, 2844, 3696, 3840, 2304, 1024

Offset

0,3

Links

Alois P. Heinz, Rows n = 0..200, flattened

Formula

G.f.: 1+t*z(2+z)/(1-2*t*z-z^2-t*z^2).

Sum_{k=0..n} (-1)^k * T(n,k) = A110164(n). - Alois P. Heinz, Apr 21 2026

Example

T(4,2) = 9 because we have (1,3),(1',3),(1,3'),(1',3'),(3,1),(3',1),(3,1'),(3',1') and (2,2).
Triangle begins:
  1;
  0, 2;
  0, 1, 4;
  0, 2, 4,  8;
  0, 1, 9, 12, 16;
  0, 2, 8, 30, 32, 32;
  ...

Maple

G:=1+t*z*(2+z)/(1-2*t*z-z^2-t*z^2): Gser:=simplify(series(G, z, 14)): for n from 0 to 12 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 12 do seq(coeff(P[n], t, k), k=0..n) od; # yields sequence in triangular form
# Alternative:
b:= proc(n) option remember; `if`(n=0, 1, expand(
      add(b(n-j)*`if`(j::odd, 2, 1)*x, j=1..n)))
    end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 21 2026

Crossrefs

Row sums yield A052945.

Alternating row sums give A110164.

Main diagonal gives A000079.

Sequence in context: A230747 A308628 A181670 * A261251 A341101 A342909

Adjacent sequences: A105471 A105472 A105473 * A105475 A105476 A105477

Keyword

nonn,tabl

Author

Emeric Deutsch, Apr 09 2005

Extensions

Row n=0 and column k=0 from Alois P. Heinz, Apr 21 2026

Status

approved