cp's OEIS Frontend

A211608 Triangle T(n,k), 0 <= k <= n, given by (0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ...) DELTA (1, 2, 3, 4, 5, 6, 7, 8, 9, ...) where DELTA is the operator defined in A084938.

%I A211608 #14 Jan 18 2018 13:22:45
%S A211608 1,0,1,0,1,3,0,1,9,15,0,1,21,90,105,0,1,45,375,1050,945,0,1,93,1350,
%T A211608 6825,14175,10395,0,1,189,4515,36750,132300,218295,135135,0,1,381,
%U A211608 14490,178605,992250,2765070,3783780,2027025
%N A211608 Triangle T(n,k), 0 <= k <= n, given by (0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ...) DELTA (1, 2, 3, 4, 5, 6, 7, 8, 9, ...) where DELTA is the operator defined in A084938.
%H A211608 K. N. Boyadzhiev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Boyadzhiev/boyadzhiev6.html">Series with central binomial coefficients, Catalan numbers, and harmonic numbers</a>, J. Integer Seq. 15 (2012), Article 12.1.7.
%F A211608 T(n,k) = A048993(n,k)*A001147(k).
%F A211608 T(n,k) = A211402(n,k)/(2^(n-k)).
%F A211608 T(n,k) = k*T(n-1,k) + (2*k-1)*T(n-1,k-1), T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n.
%F A211608 G.f.: F(x,t) = 1 + x*t + (x+3*x^2)*t^2/2! + (x+9*x^2+15*x^3)*t^3/3! + ... = Sum_{n = 0..inf} R(n,x)* t^n/n!.
%F A211608 The row polynomials R(n,x) satisfy the recursion R(n+1,x) = (x+2*x^2)*R'(n,x) + x*R(n,x) where ' indicates differentiation with respect to x.
%F A211608 R(n,x) = 1/sqrt(1 + 2*x)*Sum_{k >= 0} binomial(2*k,k)/2^k*k^n * x^k/(1 + 2*x)^k (see Boyadzhiev, eqn. 19). - _Peter Bala_, Jan 18 2018
%e A211608 Triangle begins :
%e A211608 1
%e A211608 0, 1
%e A211608 0, 1,  3
%e A211608 0, 1,  9,   15
%e A211608 0, 1, 21,   90,  105
%e A211608 0, 1, 45,  375, 1050,   945
%e A211608 0, 1, 93, 1350, 6825, 14175, 10395
%Y A211608 Cf. A000079, A001147, A048993, A187075, A211402, A142963
%K A211608 easy,nonn,tabl
%O A211608 0,6
%A A211608 _Philippe Deléham_, Feb 10 2013