OFFSET
1,7
COMMENTS
Row n gives the coefficients in the expansion of (1/x)*(1 - x)^n*N(n,x/(1 - x)), where N(n,x) is the n-th row polynomial for the triangle of Narayana numbers A001263.
LINKS
G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened
Michael Albert, Cheyne Homberger and Jay Pantone, Equipopularity Classes in the Separable Permutations, arXiv:1410.7312 [math.CO], 2014; see p. 13.
Wikipedia, Hypergeometric function
FORMULA
The n-th row of the triangle is generated by the coefficients of (1 - x)^(n - 1)*F(-n, 1 - n; 2; x/(1 - x)), where F(a, b ; c; z) is the ordinary hypergeometric function.
G.f.: ((1 - y - sqrt(1 - 2*y + ((1 - 2*x)*y)^2))/(2*(1 - x)*x*y). - Franck Maminirina Ramaharo, Oct 23 2018
EXAMPLE
Triangle begins
1;
1;
1, 1, -1;
1, 3, -3;
1, 6, -4, -4, 2;
1, 10, 0, -20, 10;
1, 15, 15, -55, 15, 15, -5;
1, 21, 49, -105, -35, 105, -35;
1, 28, 112, -140, -266, 364, -56, -56, 14;
1, 36, 216, -84, -882, 756, 336, -504, 126;
...
MATHEMATICA
p[x_, n_]:= p[x, n]= Sum[(Binomial[n, j]*Binomial[n, j-1]/n)*x^j*(1-x)^(n-j), {j, 1, n}]/x;
Table[CoefficientList[p[x, n], x], {n, 1, 12}]//Flatten
PROG
(Sage)
def p(n, x): return (1/(n*x))*sum( binomial(n, j)*binomial(n, j-1)*x^j*(1-x)^(n-j) for j in (1..n) )
def T(n): return ( p(n, x) ).full_simplify().coefficients(sparse=False)
[T(n) for n in (1..12)] # G. C. Greubel, Jul 14 2021
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Roger L. Bagula, Mar 09 2010
EXTENSIONS
Edited and new name by Joerg Arndt, Oct 28 2014
Comments and formula clarified by Franck Maminirina Ramaharo, Oct 23 2018
STATUS
approved