Real-rooted atlas

type_b_eulerian → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
218817
31267248147
41804647683841697
51242264081609600384024483
61728141687296015168013824046080423857
712186737525957281948800303744022579206451208560947
81656037750446126082230502452899840658022404128768010321920197613377
9119682191376034553280237646080804948480149022720015405465608360755201857945605131725123
101590489647528253355520241524096011230894080290323353604397137920038862028800185794560003715891200148070287697
11117714648473832183137380823764012800147692244480512191088640106602338304013641965568001052526182400449622835200817496064004699645934547
121531440243077744131104596482285271056641862254817280843079108608023161025986560402477155942404455353548800030492603187200117719433216001961990553600162723741209057
1311594322121751448093231684000216162762816022770074046720131947341926400465446465003520105478758088704015709760549683201531415376691200941591966515200331576403558400510117543936006103779096411363
141478296860939496886599268748802020050212544027208709079552019885563336038408828959762022400253785323151360004883472150036480063579244009881600553477535170560003088761728532480099983038611456001428329123020800246564971326084337
Technical details
Node id
n-a83326ab79ee2ecd · revision 0 · JSON
Definition source
derived:mobius_x_over_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: 1,     2;
  2: 1,     8,       8;
  3: 1,    26,      72,       48;
  4: 1,    80,     464,      768,       384;
  5: 1,   242,    2640,     8160,      9600,      3840;
  6: 1,   728,   14168,    72960,    151680,    138240,      46080;
  7: 1,  2186,   73752,   595728,   1948800,   3037440,    2257920,     645120;
  8: 1,  6560,  377504,  4612608,  22305024,  52899840,   65802240,   41287680,  10321920;
  9: 1, 19682, 1913760, 34553280, 237646080, 804948480, 1490227200, 1540546560, 836075520, 185794560;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*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[Together[((1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = ((1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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