The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A174128 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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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).
(history; published version)

#30 by Joerg Arndt at Thu Jul 15 04:37:28 EDT 2021

STATUS

reviewed

approved

#29 by Joerg Arndt at Thu Jul 15 01:55:53 EDT 2021

STATUS

proposed

reviewed

#28 by Michel Marcus at Thu Jul 15 01:45:05 EDT 2021

STATUS

editing

proposed

#27 by Michel Marcus at Thu Jul 15 01:45:01 EDT 2021

LINKS

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], (27-October-2014); see p. 13.

STATUS

proposed

editing

#26 by G. C. Greubel at Thu Jul 15 01:02:53 EDT 2021

STATUS

editing

proposed

#25 by G. C. Greubel at Thu Jul 15 01:02:44 EDT 2021

LINKS

G. C. Greubel, <a href="/A174128/b174128.txt">Rows n = 1..50 of the irregular triangle, flattened</a>

MATHEMATICA

p[x_, n_] := p[x, n]= Sum[(Binomial[n - 1, m - 1, j]*Binomial[n, m j- 1]/mn)*x^mj*(1 - x)^(n - mj), {m, j, 1, n}]/x;

Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], , x], {n, 1, 1012}]; //Flatten

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, A123202, A123217, A123221, A141720, A144387, A144400.

STATUS

approved

editing

#24 by Bruno Berselli at Tue Oct 23 11:10:00 EDT 2018

STATUS

reviewed

approved

#23 by Joerg Arndt at Tue Oct 23 10:37:41 EDT 2018

STATUS

proposed

reviewed

#22 by Franck Maminirina Ramaharo at Tue Oct 23 06:15:03 EDT 2018

STATUS

editing

proposed

Discussion

Tue Oct 23

10:37

Joerg Arndt: Yes!

#21 by Franck Maminirina Ramaharo at Tue Oct 23 06:05:38 EDT 2018

LINKS

Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergeometric_function">Hypergeometric function</a>

EXTENSIONS

Comments and formula clarified by Franck Maminirina Ramaharo, Oct 23 2018

Discussion

Tue Oct 23

06:14

Franck Maminirina Ramaharo: Slightly edit the name. @Joerg is it ok with the comments? Thanks