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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).

%I A174128 #34 Feb 08 2025 01:22:51
%S A174128 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,
%T A174128 21,49,-105,-35,105,-35,1,28,112,-140,-266,364,-56,-56,14,1,36,216,
%U A174128 -84,-882,756,336,-504,126,1,45,375,210,-2100,672,2520,-2100,210,210,-42
%N 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).
%C A174128 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.
%H A174128 G. C. Greubel, <a href="/A174128/b174128.txt">Rows n = 1..50 of the irregular triangle, flattened</a>
%H A174128 Michael Albert, Cheyne Homberger and Jay Pantone, <a href="http://arxiv.org/abs/1410.7312">Equipopularity Classes in the Separable Permutations</a>, arXiv:1410.7312 [math.CO], 2014; see p. 13.
%H A174128 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergeometric_function">Hypergeometric function</a>
%F A174128 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.
%F A174128 G.f.: (1 - y - sqrt(1 - 2*y + ((1 - 2*x)*y)^2))/(2*(1 - x)*x*y). - _Franck Maminirina Ramaharo_, Oct 23 2018
%e A174128 Triangle begins
%e A174128     1;
%e A174128     1;
%e A174128     1,  1,  -1;
%e A174128     1,  3,  -3;
%e A174128     1,  6,  -4,   -4,    2;
%e A174128     1, 10,   0,  -20,   10;
%e A174128     1, 15,  15,  -55,   15,  15,  -5;
%e A174128     1, 21,  49, -105,  -35, 105, -35;
%e A174128     1, 28, 112, -140, -266, 364, -56,  -56,  14;
%e A174128     1, 36, 216,  -84, -882, 756, 336, -504, 126;
%e A174128     ...
%t A174128 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;
%t A174128 Table[CoefficientList[p[x, n], x], {n, 1, 12}]//Flatten
%o A174128 (Sage)
%o A174128 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) )
%o A174128 def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
%o A174128 [T(n) for n in (1..12)] # _G. C. Greubel_, Jul 14 2021
%Y A174128 Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144387, A144400.
%K A174128 sign,tabf
%O A174128 1,7
%A A174128 _Roger L. Bagula_, Mar 09 2010
%E A174128 Edited and new name by _Joerg Arndt_, Oct 28 2014
%E A174128 Comments and formula clarified by _Franck Maminirina Ramaharo_, Oct 23 2018