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

Author

Colin Mallows, Aug 23 2000

Keywords

• nonn
• tabl
• easy
• nice

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

Examples

			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
		

Links

• 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.

Crossrefs

Columns are A000217, A005712, A005714, A005716.

Cf. A027907, A104027, A117569, A124302.

Programs

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 *)

PARI

T(n,k)=if(nPaul D. Hanna

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

Extensions

More terms from James Sellers, Aug 25 2000

More terms from Paul D. Hanna, Feb 26 2005