Real-rooted atlas

A099093 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(28\,n + 76\,n x + 40\,n x^{2} + 28\,n^{2} x + 48\,n^{2} x^{2} + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(140 + 632\,x + 944\,x^{2} + 480\,x^{3} - 56\,n + 100\,n x + 664\,n x^{2} + 576\,n x^{3} - 84\,n^{2} x - 56\,n^{2} x^{2} + 96\,n^{2} x^{3} - 16\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(49 + 308\,x + 704\,x^{2} + 688\,x^{3} + 240\,x^{4} + 98\,n x + 448\,n x^{2} + 648\,n x^{3} + 288\,n x^{4} + 28\,n^{2} x^{2} + 80\,n^{2} x^{3} + 48\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131215
21284108204
34546811888642565
41712340885612960648030807
5648110165540412182412312046656358668
6245749896312984918432139968010730883265924083129
793152199961653372606787212370320143233928817984223948845701739

All 15 rows

Technical details
Node id
n-900b35b672f58020 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-cbe33726be898ce0

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:      3,      12;
  2:     12,      84,      108;
  3:     45,     468,     1188,       864;
  4:    171,    2340,     8856,     12960,      6480;
  5:    648,   11016,    55404,    121824,    123120,      46656;
  6:   2457,   49896,   312984,    918432,   1399680,    1073088,     326592;
  7:   9315,  219996,  1653372,   6067872,  12370320,   14323392,    8817984,    2239488;
  8:  35316,  950940,  8328096,  36689760,  93707280,  145333440,  135209088,   69424128,  15116544;
  9: 133893, 4048380, 40482828, 208124640, 638915040, 1242729216, 1549679040, 1202605056, 529079040, 100776960;
%F Formula
The row polynomials satisfy (28*n+(28*n^2+76*n)*x+(8*n^3+48*n^2+40*n)*x^2)*P(n,x) = -(56*n-140+(84*n^2-100*n-632)*x+(16*n^3+56*n^2-664*n-944)*x^2-(96*n^2+576*n+480)*x^3)*P(n-1,x) + (49+(98*n+308)*x+(28*n^2+448*n+704)*x^2+(80*n^2+648*n+688)*x^3+(48*n^2+288*n+240)*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[(-(56*n - 140 + (84*n^2 - 100*n - 632)*x + (16*n^3 + 56*n^2 - 664*n - 944)*x^2 - (96*n^2 + 576*n + 480)*x^3)*P[n - 1, x] + (49 + (98*n + 308)*x + (28*n^2 + 448*n + 704)*x^2 + (80*n^2 + 648*n + 688)*x^3 + (48*n^2 + 288*n + 240)*x^4)*D[P[n - 1, x], x])/(28*n + (28*n^2 + 76*n)*x + (8*n^3 + 48*n^2 + 40*n)*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] = (-(56*n - 140 + (84*n^2 - 100*n - 632)*x + (16*n^3 + 56*n^2 - 664*n - 944)*x^2 - (96*n^2 + 576*n + 480)*x^3)*v[i - 1] + (49 + (98*n + 308)*x + (28*n^2 + 448*n + 704)*x^2 + (80*n^2 + 648*n + 688)*x^3 + (48*n^2 + 288*n + 240)*x^4)*deriv(v[i - 1], x))/(28*n + (28*n^2 + 76*n)*x + (8*n^3 + 48*n^2 + 40*n)*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.