A056241 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 19, 10, 1, 1, 15, 45, 45, 15, 1, 1, 21, 90, 141, 90, 21, 1, 1, 28, 161, 357, 357, 161, 28, 1, 1, 36, 266, 784, 1107, 784, 266, 36, 1, 1, 45, 414, 1554, 2907, 2907, 1554, 414, 45, 1, 1, 55, 615, 2850, 6765, 8953, 6765, 2850, 615, 55

Offset

1,5

Comments

Forms the even-indexed trinomial coefficients (A027907). Matrix inverse is A104027. - Paul D. Hanna, Feb 26 2005

Subtriangle (for 1<=k<=n) of triangle defined by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 29 2006

Links

Table of n, a(n) for n=1..65.

Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 6.

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 7.

F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.

Formula

T(n, k) = Sum_{j=0..k-1} C(n-1, 2k-j-2)*C(2k-j-2, j).

G.f.: A(x, y) = (1 - x*(1+y))/(1 - 2*x*(1+y) + x^2*(1+y+y^2)) (offset=0). - Paul D. Hanna, Feb 26 2005

Sum_{k, 1<=k<=n}T(n,k)=A124302(n). Sum_{k, 1<=k<=n}(-1)^(n-k)*T(n,k)=A117569(n). - Philippe Deléham, Oct 29 2006

From Paul Barry, Sep 28 2010: (Start)

G.f.: 1/(1-x-xy-x^2y/(1-x-xy)).

E.g.f.: exp((1+y)x)*cosh(sqrt(y)*x).

T(n,k) = Sum_{j=0..n} C(n,j)*C(n-j,2*(k-j)). (End)

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) - T(n-2,k-2), T(1,1) = T(2,1) = T(2,2) = 1, T(n,k) = 0 if k<1 or if k>n. - Philippe Deléham, Mar 27 2014

Example

Triangle begins:
  1;
  1,1;
  1,3,1;
  1,6,6,1;
  1,10,19,10,1;
  ...
Triangle (0, 1, 0, 1, 0, 0, 0...) DELTA (1, 0, 1, 0, 0, 0, ...) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 3, 1;
  0, 1, 6, 6, 1;
  0, 1, 10, 19, 10, 1;
  0, 1, 15, 45, 45, 15, 1;
  0, 1, 21, 90, 141, 90, 21, 1;
  ... - Philippe Deléham, Mar 27 2014

Mathematica

t[n_, k_] := Sum[ Binomial[n, j]*Binomial[n-j, 2*(k-j)], {j, 0, n}]; Flatten[ Table[t[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, Oct 11 2011, after Paul Barry *)

Prog

(PARI) T(n, k)=if(n<k || k<1, 0, polcoeff((1+x+x^2)^(n-1)+O(x^(2*k)), 2*k-2)) \\ Paul D. Hanna

Crossrefs

Columns are A000217, A005712, A005714, A005716.

Cf. A027907, A104027, A117569, A124302.

Sequence in context: A299146 A362242 A114176 * A162745 A001263 A162747

Adjacent sequences: A056238 A056239 A056240 * A056242 A056243 A056244

Keyword

nonn,tabl,easy,nice

Author

Colin Mallows, Aug 23 2000

Extensions

More terms from James A. Sellers, Aug 25 2000

More terms from Paul D. Hanna, Feb 26 2005

Status

approved