0,3

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.

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

T is the convolution triangle of the positive integers starting at 2 (see A357368). - Peter Luschny, Oct 19 2022

John Tyler Rascoe, Rows n = 0..100, flattened

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).

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

Sum_{k=0..n} k*T(n,k) = A181290(n).

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

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;

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

# Uses function PMatrix from A357368.

PMatrix(10, n -> n + 1); # Peter Luschny, Oct 19 2022

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 *)

(PARI)

T_xt(max_row) = {my(N=max_row+1, x='x+O('x^N), h=(1-x)^2/((1-x)^2 - t*x*(2-x))); vector(N, n, Vecrev(polcoeff(h, n-1)))}

T_xt(10) \\ John Tyler Rascoe, Apr 05 2025

Cf. A003480 (row sums), A181290.

Cf. A253722, A033282, A001003, A133437.

Cf. A086810, A132440, A145271, A218272.

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

Sequence in context: A368090 A383353 A209705 * A229032 A352835 A349953

Adjacent sequences: A181286 A181287 A181288 * A181290 A181291 A181292

nonn,tabl,easy

Emeric Deutsch, Oct 12 2010

approved