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
Examples
1; 1,1; 3,2,1; 9,7,3,1; 31,24,12,4,1; ...
Programs
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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
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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 *) -
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 */
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
Extensions
More terms from Emeric Deutsch, Jun 06 2005
Comments