A181301 -id:A181301 - cp's OEIS frontend

A060086 Convolution triangle A059594 with extra first column.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 3, 8, 9, 4, 1, 0, 3, 14, 19, 14, 5, 1, 0, 4, 20, 39, 36, 20, 6, 1, 0, 4, 30, 69, 85, 60, 27, 7, 1, 0, 5, 40, 119, 176, 160, 92, 35, 8, 1, 0, 5, 55, 189, 344, 376, 273, 133, 44, 9

Offset: 0

Wolfdieter Lang, Apr 06 2001

Riordan array (1, x/((1+x)*(1-x)^2)). - Philippe Deléham, Feb 24 2012

Triangle, read by rows, given by (0, 1, 1, -2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 24 2012

			{1}; {0,1}; {0,1,1}; {0,2,2,1}; ...
Triangle begins :
1
0, 1
0, 1, 1
0, 2, 2, 1
0, 2, 5, 3, 1
0, 3, 8, 9, 4, 1
0, 3, 14, 19, 14, 5, 1

Cf. A059594,

t[0, 0] = 1; t[, 0] = 0; t[n, m_] := Sum[ Sum[ Binomial[j, 2*j-3*k-m+n]*(-1)^(j-k)*Binomial[k, j], {j, 0, k}]*Binomial[m+k-1, m-1], {k, 0, n-m}]; Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

G.f.for column m >= 0: (x/((1-x^2)*(1-x)))^m.
T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k) - T(n-3,k) with T(n,0) = 0^n. - Philippe Deléham, Feb 24 2012
G.f.: (1-x-x^2+x^3)/(1-x-x^2+x^3-y*x). - Philippe Deléham, Feb 24 2012
Sum_{k, 0<=k<=n} T(n,k)*2^k = A181301(n). - Philippe Deléham, Feb 24 2012

A181299 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns in which the top entry is equal to the bottom entry (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 2, 6, 1, 20, 4, 64, 17, 1, 206, 68, 6, 662, 261, 32, 1, 2128, 976, 152, 8, 6840, 3577, 675, 51, 1, 21986, 12912, 2860, 280, 10, 70670, 46049, 11704, 1406, 74, 1, 227156, 162628, 46632, 6632, 460, 12, 730152, 569705, 181877, 29866, 2570, 101, 1, 2346942

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.

Row n contains 1+floor(n/2) entries.

			T(3,1) = 4 because we have (1,1/1,0), (1,0/1,1), (1,1/0,1), (0,1/1,1) (the 2-compositions are written as (top row/bottom row)).
Triangle starts:
  1;
  2;
  6,1;
  20,4;
  64,17,1;

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.

Cf. A003480, A181300, A181301.

G := (1-z)^2*(1+z)/(1-3*z-z^2+z^3-t*z^2*(1-z)): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

G.f.: G(t,z) = (1+z)*(1-z)^2/(1-3*z-z^2+z^3-t*(1-z)*z^2).
Sum_{k>=0} T(n,k) = A003480(n).
T(n,0) = A181301(n).
Sum_{k>=0} k*T(n,k) = A181300(n).

cp's OEIS Frontend

A060086 Convolution triangle A059594 with extra first column.

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