Real-rooted atlas

A128099 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(36\,n + 52\,n x + 36\,n^{2} x + 16\,n^{2} x^{2} + 16\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-144 - 640\,x - 832\,x^{2} - 256\,x^{3} + 180\,n + 476\,n x + 192\,n x^{2} - 128\,n x^{3} + 252\,n^{2} x + 592\,n^{2} x^{2} + 256\,n^{2} x^{3} + 80\,n^{3} x^{2} + 128\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-81 - 540\,x - 1316\,x^{2} - 1376\,x^{3} - 512\,x^{4} - 162\,n x - 828\,n x^{2} - 1392\,n x^{3} - 768\,n x^{4} - 72\,n^{2} x^{2} - 272\,n^{2} x^{3} - 256\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
23811
351621
4115648115
521128144293
6433285762561203
785752163210243493
8171175247524608128012563
934139361252815360640038565
1068388083254449920320006144130099
111365194088121614745612160036864407909
12273142488199152420864435200202752286721331859
13546192224478032114892814156808601602007044201189
14109231990481131264305280043904003354624120422413107213474355
Technical details
Node id
n-c823c2082410ea42 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-0631f52eb5bb8a38

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:   3,    8;
  3:   5,   16;
  4:  11,   56,    48;
  5:  21,  128,   144;
  6:  43,  328,   576,   256;
  7:  85,  752,  1632,  1024;
  8: 171, 1752,  4752,  4608, 1280;
  9: 341, 3936, 12528, 15360, 6400;
%F Formula
The row polynomials satisfy (36*n+(36*n^2+52*n)*x+(16*n^3+16*n^2)*x^2)*P(n,x) = (180*n-144+(252*n^2+476*n-640)*x+(80*n^3+592*n^2+192*n-832)*x^2+(128*n^3+256*n^2-128*n-256)*x^3)*P(n-1,x) - (81+(162*n+540)*x+(72*n^2+828*n+1316)*x^2+(272*n^2+1392*n+1376)*x^3+(256*n^2+768*n+512)*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[((180*n - 144 + (252*n^2 + 476*n - 640)*x + (80*n^3 + 592*n^2 + 192*n - 832)*x^2 + (128*n^3 + 256*n^2 - 128*n - 256)*x^3)*P[n - 1, x] - (81 + (162*n + 540)*x + (72*n^2 + 828*n + 1316)*x^2 + (272*n^2 + 1392*n + 1376)*x^3 + (256*n^2 + 768*n + 512)*x^4)*D[P[n - 1, x], x])/(36*n + (36*n^2 + 52*n)*x + (16*n^3 + 16*n^2)*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] = ((180*n - 144 + (252*n^2 + 476*n - 640)*x + (80*n^3 + 592*n^2 + 192*n - 832)*x^2 + (128*n^3 + 256*n^2 - 128*n - 256)*x^3)*v[i - 1] - (81 + (162*n + 540)*x + (72*n^2 + 828*n + 1316)*x^2 + (272*n^2 + 1392*n + 1376)*x^3 + (256*n^2 + 768*n + 512)*x^4)*deriv(v[i - 1], x))/(36*n + (36*n^2 + 52*n)*x + (16*n^3 + 16*n^2)*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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.