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A181289 Triangle read by rows: T(n,k) is the number of 2-compositions of n having length k (0 <= k <= n). 10
1, 0, 2, 0, 3, 4, 0, 4, 12, 8, 0, 5, 25, 36, 16, 0, 6, 44, 102, 96, 32, 0, 7, 70, 231, 344, 240, 64, 0, 8, 104, 456, 952, 1040, 576, 128, 0, 9, 147, 819, 2241, 3400, 2928, 1344, 256, 0, 10, 200, 1372, 4712, 9290, 11040, 7840, 3072, 512, 0, 11, 264, 2178, 9108, 22363 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n. The length of the 2-composition is the number of columns.

The sum of entries in row n is A003480(n). Sum_{k=0..n} k*T(n,k) = A181290(n).

From Tom Copeland, Sep 06 2011: (Start)

R(t,z) = (1-z)^2 / ((1+t)*(1-z)^2-1) = 1/(t - (2*z + 3*z^2 + 4*z^3 + 5*z^4 + ...)) = 1/t + (1/t)^2*2*z + (1/t)^3*(4+3t)*z^2 + (1/t)^4*(8+12*t+4*t^2)*z^3 + ... gives row reversed polynomials of A181289 with G(t,z) = R(1/t,z)/t.

R(t,z) is related to generators for A033282 and A001003 (t=1) and can be umbrally extended to give a partition generator for A133437. (End)

A refined, reverse version of this array is given in A253722. - Tom Copeland, May 02 2015

The infinitesimal generator (infinigen) for the face polynomials of associahedra A086810/A033282, read as decreasing powers, (and for the dual simplicial complex read as increasing powers) can be formed from the row polynomials P(n,t) of this entry. This type of infinigen is presented in A145271 for general sets of binomial Sheffer polynomials. This specific infinigen is presented in analytic form in A086810. Given the column vector of row polynomials V = (P(0,t) = 1, P(1,y) = 2 t, P(2,y) = 3 t + 4 t^2, P(3,y) = 4 t + 12 t^2 + 8 t^3, ...), form the lower triangular matrix M(n,k) = V(n-k,n-k), i.e., diagonally multiply the matrix with all ones on the diagonal and below by the components of V. Form the matrix MD by multiplying A132440^Transpose = A218272 = D (representing derivation of o.g.f.s) by M, i.e., MD = M*D. The non-vanishing component of the first row of (MD)^n * V / (n+1)! is the n-th face polynomial. - Tom Copeland, Dec 11 2015

LINKS

Table of n, a(n) for n=0..60.

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (11).

FORMULA

T(n,k) = Sum_{j=0..k} (-1)^j*2^(k-j)*binomial(k,j)*binomial(n+k-j-1, 2k-1) (0 <= k <= n).

G.f.: G(t,x) = (1-x)^2/((1-x)^2 - t*x(2-x)).

G.f. of column k = x^k*(2-x)^k/(1-x)^{2k} (k>=1) (we have a Riordan array).

Recurrences satisfied by the numbers u_{n,k}=T(n,k) can be found in the Castiglione et al. reference.

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0)=1, T(1,0)=0, T(1,1)=2, T(2,0)=0, T(1,1)=3, T(2,2)=4, T(n,k)=0, if k < 0 or if k > n. - Philippe Deléham, Nov 29 2013

EXAMPLE

Triangle starts:

  1;

  0,  2;

  0,  3,   4;

  0,  4,  12,    8;

  0,  5,  25,   36,   16;

  0,  6,  44,  102,   96,    32;

  0,  7,  70,  231,  344,   240,    64;

  0,  8, 104,  456,  952,  1040,   576,   128;

  0,  9, 147,  819, 2241,  3400,  2928,  1344,   256;

  0, 10, 200, 1372, 4712,  9290, 11040,  7840,  3072,  512;

  0, 11, 264, 2178, 9108, 22363, 34332, 33488, 20224, 6912, 1024;

MAPLE

T := proc (n, k) if k <= n then sum((-1)^j*2^(k-j)*binomial(k, j)*binomial(n+k-j-1, 2*k-1), j = 0 .. k) else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form

MATHEMATICA

Table[Sum[(-1)^j*2^(k - j) Binomial[k, j] Binomial[n + k - j - 1, 2 k - 1], {j, 0, k}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 11 2015 *)

CROSSREFS

Cf. A003480, A181290.

Cf. A253722, A033282, A001003, A133437.

Cf. A086810, A132440, A145271, A218272.

Cf. A000297 (column 3), A006636 (column 4), A006637 (column 5).

Sequence in context: A227595 A078436 A209705 * A229032 A117909 A261094

Adjacent sequences:  A181286 A181287 A181288 * A181290 A181291 A181292

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Oct 12 2010

STATUS

approved

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Last modified January 17 20:36 EST 2020. Contains 330987 sequences. (Running on oeis4.)