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A109971 Inverse of Riordan array (1,x(1-x)^2), A109970.

%I A109971 #15 Jun 28 2017 07:51:27
%S A109971 1,0,1,0,2,1,0,7,4,1,0,30,18,6,1,0,143,88,33,8,1,0,728,455,182,52,10,
%T A109971 1,0,3876,2448,1020,320,75,12,1,0,21318,13566,5814,1938,510,102,14,1,
%U A109971 0,120175,76912,33649,11704,3325,760,133,16,1,0,690690,444015,197340
%N A109971 Inverse of Riordan array (1,x(1-x)^2), A109970.
%C A109971 Row sums are A001764. Diagonal sums are A109972. Second column is A006013. Third column is A006629.
%H A109971 Naiomi Cameron, J. E. McLeod, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/McLeod/mcleod3.html">Returns and Hills on Generalized Dyck Paths</a>, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
%H A109971 W.-j. Woan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Woan/woan655.html">The Lagrange inversion formula and divisibility properties</a>, JIS 10 (2007) 07.7.8, example 4.
%F A109971 Number triangle T(0, 0)=1, T(0, k)=0, k>0, T(n, k)=(k/n)*binomial(3n-k-1, n-k) otherwise; Riordan array (1, f) where f(1-f)^2=x.
%F A109971 T(n, k)=sum{j=0..n, ((3j+1)/(2n+j+1))(-1)^(j-k)*C(3n, 2n+j)C(j, k)}; - _Paul Barry_, Oct 07 2005
%F A109971 T(n,k)=binomial(3n-k,n-k)*2k/(3n-k). (Paul Barry, May 18 2006)
%e A109971 Rows begin
%e A109971 1;
%e A109971 0,1;
%e A109971 0,2,1;
%e A109971 0,7,4,1;
%e A109971 0,30,18,6,1;
%e A109971 0,143,88,33,8,1;
%e A109971 Production array begins
%e A109971 0, 1
%e A109971 0, 2, 1
%e A109971 0, 3, 2, 1
%e A109971 0, 4, 3, 2, 1
%e A109971 0, 5, 4, 3, 2, 1
%e A109971 0, 6, 5, 4, 3, 2, 1,
%e A109971 0, 7, 6, 5, 4, 3, 2, 1
%e A109971 0, 8, 7, 6, 5, 4, 3, 2, 1
%e A109971 0, 9, 8, 7, 6, 5, 4, 3, 2, 1
%e A109971 ... - _Philippe Deléham_, Mar 05 2013
%Y A109971 Essentially the same as A092276.
%K A109971 easy,nonn,tabl
%O A109971 0,5
%A A109971 _Paul Barry_, Jul 06 2005