Real-rooted atlas

A113413 → basis A013610 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(9 + 9\,n\right)\,P_{n}(x) = \left(225 + 108\,x - 135\,n - 54\,n x + 18\,n^{2}\right)\,P_{n-1}(x) + \left(498 + 549\,x + 135\,x^{2} - 123\,n - 126\,n x - 27\,n x^{2}\right)\,P_{n-1}'(x) + \left(164 + 291\,x + 162\,x^{2} + 27\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
23436979
319431216227695
4110623881890648816113
5630617088181989504243024353769
635954117228156978110376429308748729472943
7204994781272126063011154245840101807923061821874159927

All 15 rows

Technical details
Node id
n-ecfc5d801b62ba29 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-cde29bc77d3d16a4

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:       6,        3;
  2:      34,       36,        9;
  3:     194,      312,      162,       27;
  4:    1106,     2388,     1890,      648,       81;
  5:    6306,    17088,    18198,     9504,     2430,      243;
  6:   35954,   117228,   156978,   110376,    42930,     8748,      729;
  7:  204994,   781272,  1260630,  1115424,   584010,   180792,    30618,    2187;
  8: 1168786,  5098116,  9626562, 10271448,  6769170,  2819772,   724626,  104976,   6561;
  9: 6663906, 32737008, 70813494, 88480512, 70342506, 36881568, 12747294, 2799360, 354294, 19683;
%F Formula
The row polynomials satisfy (9*n+9)*P(n,x) = (18*n^2-135*n+225-(54*n-108)*x)*P(n-1,x) - (123*n-498+(126*n-549)*x+(27*n-135)*x^2)*d/dx P(n-1,x) + (164+291*x+162*x^2+27*x^3)*d^2/dx^2 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[((18*n^2 - 135*n + 225 - (54*n - 108)*x)*P[n - 1, x] - (123*n - 498 + (126*n - 549)*x + (27*n - 135)*x^2)*D[P[n - 1, x], x] + (164 + 291*x + 162*x^2 + 27*x^3)*D[P[n - 1, x], {x, 2}])/(9*n + 9)]];
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] = ((18*n^2 - 135*n + 225 - (54*n - 108)*x)*v[i - 1] - (123*n - 498 + (126*n - 549)*x + (27*n - 135)*x^2)*deriv(v[i - 1], x) + (164 + 291*x + 162*x^2 + 27*x^3)*deriv(deriv(v[i - 1], x), x))/(9*n + 9)); 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

Candidate equalities ?

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