Real-rooted atlas

type_b_eulerian → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
288117
34872261147
43847684648011697
53840960081602640242124483
64608013824015168072960141687281423857
764512022579203037440194880059572873752218618560947
81032192041287680658022405289984022305024461260837750465601197613377
9185794560836075520154054656014902272008049484802376460803455328019137601968215131725123
103715891200185794560003886202880043971379200290323353601123089408024152409602533555209647528590481148070287697
11817496064004496228352001052526182400136419655680010660233830405121910886401476922444802376401280018313738084847383217714614699645934547
121961990553600117719433216003049260318720044553535488000402477155942402316102598656084307910860801862254817280228527105664131104596482430777445314401162723741209057
1351011754393600331576403558400941591966515200153141537669120015709760549683201054787580887040465446465003520131947341926400227700740467202161627628160932316840001217514480159432216103779096411363
141428329123020800999830386114560030887617285324800553477535170560006357924400988160048834721500364800253785323151360008828959762022400198855633360384027208709079552020200502125440659926874880609394968847829681246564971326084337
Technical details
Node id
n-51f13120e40d8473 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2a2e8db5a042c827

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         2,         1;
  2:         8,         8,          1;
  3:        48,        72,         26,          1;
  4:       384,       768,        464,         80,         1;
  5:      3840,      9600,       8160,       2640,       242,         1;
  6:     46080,    138240,     151680,      72960,     14168,       728,        1;
  7:    645120,   2257920,    3037440,    1948800,    595728,     73752,     2186,       1;
  8:  10321920,  41287680,   65802240,   52899840,  22305024,   4612608,   377504,    6560,     1;
  9: 185794560, 836075520, 1540546560, 1490227200, 804948480, 237646080, 34553280, 1913760, 19682, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+(2*n-1)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + (2*n - 1)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + (2*n - 1)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003