Real-rooted atlas

A241171 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 30\,x - 90\,x^{2}\right)\,P_{n-2}(x) + \left(-5\,x - 39\,x^{2} - 78\,x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} - 8\,x^{3} - 12\,x^{4}\right)\,P_{n-2}''(x) + \left(5 + 30\,x\right)\,P_{n-1}(x) + \left(5\,x + 15\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
166
230180210
31262520756010206
451026460226800453600707370
520462534404740120249480003742200067365606
68190234234086486400908107200340540200040864824008488828530
732766213267601482160680278918640001975133160005721075360005721075360001371123772206

All 15 rows

Technical details
Node id
n-59467d32d5647a53 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c1fcefc8c9c99a70

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      0;
  1:      6;
  2:     30,        180;
  3:    126,       2520,         7560;
  4:    510,      26460,       226800,         453600;
  5:   2046,     253440,      4740120,       24948000,        37422000;
  6:   8190,    2342340,     86486400,      908107200,      3405402000,       4086482400;
  7:  32766,   21326760,   1482160680,    27891864000,    197513316000,     572107536000,      572107536000;
  8: 131070,  192923820,  24610294800,   786226240800,   9411558156000,   49601723371200,   116709937344000,   100037089152000;
  9: 524286, 1740246480, 401867517240, 21132273708000, 403679758482000, 3458226595824000, 14413677261984000, 28510570408320000, 21382927806240000;
%F Formula
The row polynomials satisfy P(n,x) = -(4+30*x+90*x^2)*P(n-2,x) - (5*x+39*x^2+78*x^3)*d/dx P(n-2,x) - (x^2+8*x^3+12*x^4)*d^2/dx^2 P(n-2,x) + (5+30*x)*P(n-1,x) + (5*x+15*x^2)*d/dx P(n-1,x) + (x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[-(4 + 30*x + 90*x^2)*P[n - 2, x] - (5*x + 39*x^2 + 78*x^3)*D[P[n - 2, x], x] - (x^2 + 8*x^3 + 12*x^4)*D[P[n - 2, x], {x, 2}] + (5 + 30*x)*P[n - 1, x] + (5*x + 15*x^2)*D[P[n - 1, x], x] + (x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = -(4 + 30*x + 90*x^2)*v[i - 2] - (5*x + 39*x^2 + 78*x^3)*deriv(v[i - 2], x) - (x^2 + 8*x^3 + 12*x^4)*deriv(deriv(v[i - 2], x), x) + (5 + 30*x)*v[i - 1] + (5*x + 15*x^2)*deriv(v[i - 1], x) + (x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.