Real-rooted atlas

A114687 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 36\,x + 3\,n + 54\,n x + 54\,n x^{2} + 18\,n^{2} x + 81\,n^{2} x^{2} + 27\,n^{3} x^{2}\right)\,P_{n}(x) = \left(6 + 36\,x + 21\,n + 252\,n x + 702\,n x^{2} + 324\,n x^{3} + 144\,n^{2} x + 891\,n^{2} x^{2} + 1134\,n^{2} x^{3} + 189\,n^{3} x^{2} + 486\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1 - 21\,x - 144\,x^{2} - 378\,x^{3} - 324\,x^{4} - 12\,n x - 153\,n x^{2} - 621\,n x^{3} - 810\,n x^{4} - 18\,n^{2} x^{2} - 162\,n^{2} x^{3} - 324\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131215
21184108203
34550412968642709
419729041123215552648036365
5903165008532018144016200046656492819
64279932766042601740960243000015396483265926739015
720793526512410054414938560285768002874009613716864223948892859657

All 15 rows

Technical details
Node id
n-9c70e938e6cc3ad2 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8b1be86d811bfd46

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:     11,       84,       108;
  3:     45,      504,      1296,       864;
  4:    197,     2904,     11232,     15552,       6480;
  5:    903,    16500,     85320,    181440,     162000,      46656;
  6:   4279,    93276,    604260,   1740960,    2430000,    1539648,     326592;
  7:  20793,   526512,   4100544,  14938560,   28576800,   28740096,   13716864,    2239488;
  8: 103049,  2972208,  27046656, 119266560,  290848320,  404974080,  310915584,  116453376,  15116544;
  9: 518859, 16791516, 174847248, 905603328, 2689485120, 4808740608, 5157540864, 3144241152, 952342272, 100776960;
%F Formula
The row polynomials satisfy (3*n+6+(18*n^2+54*n+36)*x+(27*n^3+81*n^2+54*n)*x^2)*P(n,x) = (21*n+6+(144*n^2+252*n+36)*x+(189*n^3+891*n^2+702*n)*x^2+(486*n^3+1134*n^2+324*n)*x^3)*P(n-1,x) - (1+(12*n+21)*x+(18*n^2+153*n+144)*x^2+(162*n^2+621*n+378)*x^3+(324*n^2+810*n+324)*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[((21*n + 6 + (144*n^2 + 252*n + 36)*x + (189*n^3 + 891*n^2 + 702*n)*x^2 + (486*n^3 + 1134*n^2 + 324*n)*x^3)*P[n - 1, x] - (1 + (12*n + 21)*x + (18*n^2 + 153*n + 144)*x^2 + (162*n^2 + 621*n + 378)*x^3 + (324*n^2 + 810*n + 324)*x^4)*D[P[n - 1, x], x])/(3*n + 6 + (18*n^2 + 54*n + 36)*x + (27*n^3 + 81*n^2 + 54*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] = ((21*n + 6 + (144*n^2 + 252*n + 36)*x + (189*n^3 + 891*n^2 + 702*n)*x^2 + (486*n^3 + 1134*n^2 + 324*n)*x^3)*v[i - 1] - (1 + (12*n + 21)*x + (18*n^2 + 153*n + 144)*x^2 + (162*n^2 + 621*n + 378)*x^3 + (324*n^2 + 810*n + 324)*x^4)*deriv(v[i - 1], x))/(3*n + 6 + (18*n^2 + 54*n + 36)*x + (27*n^3 + 81*n^2 + 54*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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023