Real-rooted atlas

A113953 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
2912324
32763335128
4812702529011704
524310531458846219213840
672938887209604825475284320992
72187138513231936207216817137122185114688
865614811413559419197015195670470190622790171626688
9196831640255423769331209297185658662138404903262553413424256
1059049551124209295942515285145282392299219341186162481226611388468318710528
11177147183051978535171842984926370846243588061497997861976341706103299907304891365102236160
125314416022998288159127680306612723747313882201210312725653023572188816674574286720000664502731558628864
131594323196633171038081423100072505848016137387882836470983084007954521768143335523641111945934170209814381154613052404736
144782969637729203682886131218771360258286230937181974323767490225274589222414556343955618607121562832633052080039722313094801092316678649856
Technical details
Node id
n-950dddc8219098b4 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-354f21f96faadf4a

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:     3,      1;
  2:     9,     12,      3;
  3:    27,     63,     33,      5;
  4:    81,    270,    252,     90,     11;
  5:   243,   1053,   1458,    846,    219,     21;
  6:   729,   3888,   7209,   6048,   2547,    528,     43;
  7:  2187,  13851,  32319,  36207,  21681,   7137,   1221,    85;
  8:  6561,  48114, 135594, 191970, 151956,  70470,  19062,  2790,  171;
  9: 19683, 164025, 542376, 933120, 929718, 565866, 213840, 49032, 6255, 341;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n+(6*n-5)*x+(3*n-3)*x^2)*P(n-1,x) - (3*x+10*x^2+3*x^3)*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[((3*n + (6*n - 5)*x + (3*n - 3)*x^2)*P[n - 1, x] - (3*x + 10*x^2 + 3*x^3)*D[P[n - 1, x], x])/(n)]];
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] = ((3*n + (6*n - 5)*x + (3*n - 3)*x^2)*v[i - 1] - (3*x + 10*x^2 + 3*x^3)*deriv(v[i - 1], x))/(n)); 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

Same fingerprint ?

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