Real-rooted atlas

A099040 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n - 22\,n x - 24\,n x^{2} + 18\,n x^{3} - 3\,n^{2} + 6\,n^{2} x + 18\,n^{2} x^{2} + 3\,n^{3}\right)\,P_{n}(x) = \left(-28 - 136\,x - 156\,x^{2} + 36\,x^{3} + 108\,x^{4} - 40\,n - 58\,n x + 48\,n x^{2} + 72\,n x^{3} + 42\,n^{2} + 42\,n^{2} x - 36\,n^{2} x^{2} - 6\,n^{3}\right)\,P_{n-1}(x) + \left(-4 - 24\,x - 20\,x^{2} + 108\,x^{3} + 216\,x^{4} + 108\,x^{5} - 6\,n - 12\,n x + 66\,n x^{2} + 180\,n x^{3} + 108\,n x^{4} + 6\,n^{2} + 24\,n^{2} x + 18\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18614
23610236174
31366969362161984
4476352886767560129621536
51584154325558488776570247776226176
6510461656293472689688811296412128466562320000
7160642317441373184427852874183046889536289267227993623379968

All 15 rows

Technical details
Node id
n-d25eb2031e5860a1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8b252c0d2ddb189d

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:      8,       6;
  2:     36,     102,       36;
  3:    136,     696,      936,       216;
  4:    476,    3528,     8676,      7560,      1296;
  5:   1584,   15432,    55584,     88776,     57024,      7776;
  6:   5104,   61656,   293472,    689688,    811296,    412128,     46656;
  7:  16064,  231744,  1373184,   4278528,   7418304,   6889536,   2892672,    279936;
  8:  49680,  833472,  5914080,  22996224,  52955856,  72518976,  55567296,  19875456,   1679616;
  9: 151552, 2899296, 23974272, 111844800, 321916032, 586007136, 661955328, 431381376, 134369280, 10077696;
%F Formula
The row polynomials satisfy (3*n^3-3*n^2-4*n+(6*n^2-22*n)*x+(18*n^2-24*n)*x^2+18*n*x^3)*P(n,x) = -(6*n^3-42*n^2+40*n+28-(42*n^2-58*n-136)*x+(36*n^2-48*n+156)*x^2-(72*n+36)*x^3-108*x^4)*P(n-1,x) + (6*n^2-6*n-4+(24*n^2-12*n-24)*x+(18*n^2+66*n-20)*x^2+(180*n+108)*x^3+(108*n+216)*x^4+108*x^5)*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[(-(6*n^3 - 42*n^2 + 40*n + 28 - (42*n^2 - 58*n - 136)*x + (36*n^2 - 48*n + 156)*x^2 - (72*n + 36)*x^3 - 108*x^4)*P[n - 1, x] + (6*n^2 - 6*n - 4 + (24*n^2 - 12*n - 24)*x + (18*n^2 + 66*n - 20)*x^2 + (180*n + 108)*x^3 + (108*n + 216)*x^4 + 108*x^5)*D[P[n - 1, x], x])/(3*n^3 - 3*n^2 - 4*n + (6*n^2 - 22*n)*x + (18*n^2 - 24*n)*x^2 + 18*n*x^3)]];
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] = (-(6*n^3 - 42*n^2 + 40*n + 28 - (42*n^2 - 58*n - 136)*x + (36*n^2 - 48*n + 156)*x^2 - (72*n + 36)*x^3 - 108*x^4)*v[i - 1] + (6*n^2 - 6*n - 4 + (24*n^2 - 12*n - 24)*x + (18*n^2 + 66*n - 20)*x^2 + (180*n + 108)*x^3 + (108*n + 216)*x^4 + 108*x^5)*deriv(v[i - 1], x))/(3*n^3 - 3*n^2 - 4*n + (6*n^2 - 22*n)*x + (18*n^2 - 24*n)*x^2 + 18*n*x^3)); 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. [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
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.