A174128 Irregular triangle read by rows: row n (n > 0) is the expansion of Sum_{m=1..n} A001263(n,m)*x^(m - 1)*(1 - x)^(n - m).

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, 1, 45, 375, 210, -2100, 672, 2520, -2100, 210, 210, -42

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

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144387, A144400.

Sequence in context: A016454 A065227 A208539 * A131070 A295290 A165202

Adjacent sequences: A174125 A174126 A174127 * A174129 A174130 A174131

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