Real-rooted atlas

A099097 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-156 - 288\,x + 156\,n + 132\,n x - 240\,n x^{2} + 156\,n^{2} x + 216\,n^{2} x^{2} + 24\,n^{3} x^{2}\right)\,P_{n}(x) = \left(676 + 2080\,x + 1584\,x^{2} - 104\,n + 1108\,n x + 3848\,n x^{2} + 2880\,n x^{3} - 156\,n^{2} x - 56\,n^{2} x^{2} + 288\,n^{2} x^{3} - 16\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(169 + 858\,x + 1436\,x^{2} + 792\,x^{3} + 338\,n x + 1664\,n x^{2} + 2696\,n x^{3} + 1440\,n x^{4} + 52\,n^{2} x^{2} + 176\,n^{2} x^{3} + 144\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
231215
31076108194
43337210448642313
51091628681611232648026265
63606696372609187210584046656288684
711892647218374460310410411209175683265923099789
8392710182084682834704967982160105494407511616223948832705775

All 15 rows

Technical details
Node id
n-d15f2d3bdc33e8b9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a58b871b97e9221f

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     3,      12;
   3:    10,      76,      108;
   4:    33,     372,     1044,      864;
   5:   109,    1628,     6816,    11232,      6480;
   6:   360,    6696,    37260,    91872,    105840,     46656;
   7:  1189,   26472,   183744,   603104,   1041120,    917568,    326592;
   8:  3927,  101820,   846828,  3470496,   7982160,  10549440,   7511616,   2239488;
   9: 12970,  383812,  3720024, 18279200,  52524080,  92536128,  98848512,  58973184,  15116544;
  10: 42837, 1424604, 15766668, 90326496, 311324640, 685488384, 976425408, 874146816, 448457472, 100776960;
%F Formula
The row polynomials satisfy (156*n-156+(156*n^2+132*n-288)*x+(24*n^3+216*n^2-240*n)*x^2)*P(n,x) = -(104*n-676+(156*n^2-1108*n-2080)*x+(16*n^3+56*n^2-3848*n-1584)*x^2-(288*n^2+2880*n)*x^3)*P(n-1,x) + (169+(338*n+858)*x+(52*n^2+1664*n+1436)*x^2+(176*n^2+2696*n+792)*x^3+(144*n^2+1440*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(104*n - 676 + (156*n^2 - 1108*n - 2080)*x + (16*n^3 + 56*n^2 - 3848*n - 1584)*x^2 - (288*n^2 + 2880*n)*x^3)*P[n - 1, x] + (169 + (338*n + 858)*x + (52*n^2 + 1664*n + 1436)*x^2 + (176*n^2 + 2696*n + 792)*x^3 + (144*n^2 + 1440*n)*x^4)*D[P[n - 1, x], x])/(156*n - 156 + (156*n^2 + 132*n - 288)*x + (24*n^3 + 216*n^2 - 240*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(104*n - 676 + (156*n^2 - 1108*n - 2080)*x + (16*n^3 + 56*n^2 - 3848*n - 1584)*x^2 - (288*n^2 + 2880*n)*x^3)*v[i - 1] + (169 + (338*n + 858)*x + (52*n^2 + 1664*n + 1436)*x^2 + (176*n^2 + 2696*n + 792)*x^3 + (144*n^2 + 1440*n)*x^4)*deriv(v[i - 1], x))/(156*n - 156 + (156*n^2 + 132*n - 288)*x + (24*n^3 + 216*n^2 - 240*n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.