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

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, 1, 0, 3876, 2448, 1020, 320, 75, 12, 1, 0, 21318, 13566, 5814, 1938, 510, 102, 14, 1, 0, 120175, 76912, 33649, 11704, 3325, 760, 133, 16, 1, 0, 690690, 444015, 197340

Offset

0,5

Comments

Row sums are A001764. Diagonal sums are A109972. Second column is A006013. Third column is A006629.

Links

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

Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.

W.-j. Woan, The Lagrange inversion formula and divisibility properties, JIS 10 (2007) 07.7.8, example 4.

Formula

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.

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

T(n,k)=binomial(3n-k,n-k)*2k/(3n-k). (Paul Barry, May 18 2006)

Example

Rows begin
1;
0,1;
0,2,1;
0,7,4,1;
0,30,18,6,1;
0,143,88,33,8,1;
Production array begins
0, 1
0, 2, 1
0, 3, 2, 1
0, 4, 3, 2, 1
0, 5, 4, 3, 2, 1
0, 6, 5, 4, 3, 2, 1,
0, 7, 6, 5, 4, 3, 2, 1
0, 8, 7, 6, 5, 4, 3, 2, 1
0, 9, 8, 7, 6, 5, 4, 3, 2, 1
... - Philippe Deléham, Mar 05 2013

Crossrefs

Essentially the same as A092276.

Sequence in context: A174869 A330862 A269158 * A357585 A284797 A316135

Adjacent sequences: A109968 A109969 A109970 * A109972 A109973 A109974

Keyword

easy,nonn,tabl

Author

Paul Barry, Jul 06 2005

Status

approved