Real-rooted atlas

A127952 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 10\,x + 2\,x^{2} - 4\,n - 5\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(10\,x + 4\,x^{2} - 5\,n x - 2\,n x^{2}\right)\,P_{n-2}'(x) + \left(2\,x^{2} - n x^{2}\right)\,P_{n-2}''(x) + \left(-8 - 4\,x + 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-4\,x + 2\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)
\(P_{2}(x) = 6 + 12\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2612321
31656324108
440230230655570
596876138066611463138
62243178746254951575182718123
75121115237744399281708032562728109952
81152382141823942670571602094485661113879700389
925601286608531001686310136479052689010416010660530104677670
1056324273943895166102131151086255555583221496187220110175567041132696752
11122881404888174633365995717282241940542954281912125637932844316402760091212238749756
1226624457919877196418343686629599526213500835764224537599576992138785491796510417561157131817712585
13573441482300433746986019337549144243985746442117475824738393688010482481567505941889687813973966116614421414403260732
141228804770681014621811901072097617529354645775377059382702594131039510374936000253773663039066027038213175234916587165177015118576865685
Technical details
Node id
n-fdeb1554f3b02f13 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-a546e680aefa4794

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:    2,      2;
  2:    6,     12,      3;
  3:   16,     56,     32,       4;
  4:   40,    230,    230,      65,       5;
  5:   96,    876,   1380,     666,     114,      6;
  6:  224,   3178,   7462,    5495,    1575,    182,      7;
  7:  512,  11152,  37744,   39928,   17080,   3256,    272,     8;
  8: 1152,  38214, 182394,  267057,  160209,  44856,   6111,   387,   9;
  9: 2560, 128660, 853100, 1686310, 1364790, 526890, 104160, 10660, 530, 10;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(5*n-10)*x+(n-2)*x^2)*P(n-2,x) - ((5*n-10)*x+(2*n-4)*x^2)*d/dx P(n-2,x) - (n-2)*x^2*d^2/dx^2 P(n-2,x) + (4*n-8+(2*n-4)*x)*P(n-1,x) + (2*n-4)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 6 + 12*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 6 + 12*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (5*n - 10)*x + (n - 2)*x^2)*P[n - 2, x] - ((5*n - 10)*x + (2*n - 4)*x^2)*D[P[n - 2, x], x] - (n - 2)*x^2*D[P[n - 2, x], {x, 2}] + (4*n - 8 + (2*n - 4)*x)*P[n - 1, x] + (2*n - 4)*x*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 6 + 12*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (5*n - 10)*x + (n - 2)*x^2)*v[i - 2] - ((5*n - 10)*x + (2*n - 4)*x^2)*deriv(v[i - 2], x) - (n - 2)*x^2*deriv(deriv(v[i - 2], x), x) + (4*n - 8 + (2*n - 4)*x)*v[i - 1] + (2*n - 4)*x*deriv(v[i - 1], x))/(n - 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

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.