A006636 -id:A006636 - cp's OEIS frontend

A181289 Triangle read by rows: T(n,k) is the number of 2-compositions of n having length k (0 <= k <= n).

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

Offset: 0

Emeric Deutsch, Oct 12 2010

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

			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;

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).

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).

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 *)

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

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

A006637 Expansion of (2 - x)^4/(1 - x)^8.

Original entry on oeis.org

16, 96, 344, 952, 2241, 4712, 9108, 16488, 28314, 46552, 73788, 113360, 169507, 247536, 354008, 496944, 686052, 932976, 1251568, 1658184, 2172005, 2815384, 3614220, 4598360, 5802030, 7264296, 9029556, 11148064, 13676487, 16678496, 20225392, 24396768, 29281208

Offset: 0

Simon Plouffe

Former name: From generalized Catalan numbers. - G. C. Greubel, Sep 03 2025

H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A006633, A006634, A006635, A006636, A181289.

A006637:= func< n | (n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)/5040 >;
[A006637(n): n in [0..40]]; // G. C. Greubel, Sep 03 2025

Table[(n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)/7!, {n,0,40}] (* G. C. Greubel, Sep 03 2025 *)

def A006637(n): return (n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)//5040
print([A006637(n) for n in range(41)]) # G. C. Greubel, Sep 03 2025

G.f.: (2-x)^4/(1-x)^8. - Sean A. Irvine, May 31 2017
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Wesley Ivan Hurt, Jun 18 2022
From G. C. Greubel, Sep 03 2025: (Start)
a(n) = Sum_{k=0..4} binomial(4, k)*binomial(n+k+3, k+3).
a(n) = (1/7!)*(n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2 + 31*n + 192).
E.g.f.: (1/7!)*(80640 + 403200*x + 423360*x^2 + 161280*x^3 + 27090*x^4 + 2142*x^5 + 77*x^6 + x^7)*exp(x). (End)

a(6) and a(8) corrected and more terms from Sean A. Irvine, May 31 2017

New name by G. C. Greubel, Sep 03 2025

cp's OEIS Frontend

A181289 Triangle read by rows: T(n,k) is the number of 2-compositions of n having length k (0 <= k <= n).

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