Real-rooted atlas

A345013 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + 2\,n x^{2} + 3\,n^{2} + 4\,n^{2} x + n^{2} x^{2} + 4\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(2\,x + 4\,x^{2} + 2\,x^{3} + 4\,n + 12\,n x + 14\,n x^{2} + 4\,n x^{3} + 16\,n^{2} + 23\,n^{2} x + 15\,n^{2} x^{2} + 3\,n^{2} x^{3} + 18\,n^{3} + 20\,n^{3} x + 6\,n^{3} x^{2} + 4\,n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(-2\,x - 6\,x^{2} - 6\,x^{3} - 2\,x^{4} - 8\,n x - 16\,n x^{2} - 10\,n x^{3} - 2\,n x^{4} - 9\,n^{2} x - 13\,n^{2} x^{2} - 4\,n^{2} x^{3} - 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17310
23532673
31612259010486
47141344840200153113
531027350630023803852119538
613299380164158021840567067228121105
7564851891892522521709406237011970109236744334

All 15 rows

Technical details
Node id
n-9a59678cd130a8b0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-151b5c604cd872d4

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:      7,       3;
  2:     35,      32,       6;
  3:    161,     225,      90,      10;
  4:    714,    1344,     840,     200,      15;
  5:   3102,    7350,    6300,    2380,     385,      21;
  6:  13299,   38016,   41580,   21840,    5670,     672,     28;
  7:  56485,  189189,  252252,  170940,   62370,   11970,   1092,    36;
  8: 238238,  915200, 1441440, 1201200,  570570,  155232,  23100,  1680,   45;
  9: 999362, 4332042, 7876440, 7807800, 4594590, 1639638, 348348, 41580, 2475, 55;
%F Formula
The row polynomials satisfy (n^4+4*n^3+3*n^2-2*n+(2*n^3+4*n^2)*x+(n^2+2*n)*x^2)*P(n,x) = (4*n^4+18*n^3+16*n^2+4*n+(3*n^4+20*n^3+23*n^2+12*n+2)*x+(6*n^3+15*n^2+14*n+4)*x^2+(3*n^2+4*n+2)*x^3)*P(n-1,x) - ((2*n^3+9*n^2+8*n+2)*x+(2*n^3+13*n^2+16*n+6)*x^2+(4*n^2+10*n+6)*x^3+(2*n+2)*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[((4*n^4 + 18*n^3 + 16*n^2 + 4*n + (3*n^4 + 20*n^3 + 23*n^2 + 12*n + 2)*x + (6*n^3 + 15*n^2 + 14*n + 4)*x^2 + (3*n^2 + 4*n + 2)*x^3)*P[n - 1, x] - ((2*n^3 + 9*n^2 + 8*n + 2)*x + (2*n^3 + 13*n^2 + 16*n + 6)*x^2 + (4*n^2 + 10*n + 6)*x^3 + (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 + 4*n^3 + 3*n^2 - 2*n + (2*n^3 + 4*n^2)*x + (n^2 + 2*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] = ((4*n^4 + 18*n^3 + 16*n^2 + 4*n + (3*n^4 + 20*n^3 + 23*n^2 + 12*n + 2)*x + (6*n^3 + 15*n^2 + 14*n + 4)*x^2 + (3*n^2 + 4*n + 2)*x^3)*v[i - 1] - ((2*n^3 + 9*n^2 + 8*n + 2)*x + (2*n^3 + 13*n^2 + 16*n + 6)*x^2 + (4*n^2 + 10*n + 6)*x^3 + (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 + 4*n^3 + 3*n^2 - 2*n + (2*n^3 + 4*n^2)*x + (n^2 + 2*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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.