A108073 Triangle in A071943 with rows reversed.

1, 1, 1, 3, 2, 1, 9, 7, 3, 1, 31, 24, 12, 4, 1, 113, 89, 46, 18, 5, 1, 431, 342, 183, 76, 25, 6, 1, 1697, 1355, 741, 323, 115, 33, 7, 1, 6847, 5492, 3054, 1376, 520, 164, 42, 8, 1, 28161, 22669, 12768, 5900, 2326, 786, 224, 52, 9, 1, 117631, 94962, 54033, 25464

Offset

0,4

Comments

A convolution triangle based on A052709 (with first term omitted). - Philippe Deléham, Sep 15 2005

Links

Table of n, a(n) for n=0..58.

Formula

G.f.: (1-q)/(z(2 - t + 2z + tq)), where q = sqrt(1 - 4z - 4z^2). - Emeric Deutsch, Jun 06 2005

T(0, 0) = 1; T(n, k) = 0 if k < 0 or if k > n; T(n, k) = Sum_{j>=0} T(n-1, k-1+j) + Sum_{j>=0} T(n-1, k+1+j). - Philippe Deléham, Sep 15 2005

T(n,k) = k*Sum_{i=1..(n-k)} C(i,n-k-i)*C(k+2*i-1,i)/(k+i), n > k, T(n,n)=1. - Vladimir Kruchinin, Apr 27 2015

Example

1; 1,1; 3,2,1; 9,7,3,1; 31,24,12,4,1; ...

Maple

q:=sqrt(1-4*z-4*z^2): G:=(1-q)/z/(2-t+2*z+t*q): Gserz:=simplify(series(G, z=0, 14)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gserz, z^n) od: for n from 0 to 10 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form # Emeric Deutsch, Jun 06 2005

Mathematica

T[n_, n_] = 1; T[n_, k_] := (k+1)*Sum[Binomial[i, n-k-i] * Binomial[k+2*i, i] / (k+i+1), {i, 1, n-k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Vladimir Kruchinin *)

Prog

(Maxima)
T(n, k):=if n=k then 1 else k*sum((binomial(i, n-k-i)*binomial(k+2*i-1, i))/(k+i), i, 1, n-k); /* Vladimir Kruchinin, Apr 27 2015 */

Crossrefs

Row sums yield A071356. Column 0 yields A052709.

Sequence in context: A193918 A298804 A155788 * A057731 A126074 A108916

Adjacent sequences: A108070 A108071 A108072 * A108074 A108075 A108076

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Jun 05 2005

Extensions

More terms from Emeric Deutsch, Jun 06 2005

Status

approved