Real-rooted atlas

A113953 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(4\,n\right)\,P_{n}(x) = \left(-20\,x - 9\,x^{2} + 16\,n + 24\,n x + 9\,n x^{2}\right)\,P_{n-1}(x) + \left(-16\,x - 40\,x^{2} - 9\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
21616335
364112445225
4256640448120111475
51024332834561504292219625
6409616384227841433645287044362875
716384778241361921144325139212688162885410625
8655363604487618568089604802561670403388837201712681875
92621441638400406323252428803917824178841650688087168834034117515625
1010485767340032209059843185049628909568165314566112768146073621806418512683114396875
1141943043250585610459545618409062419755827213686374463125504195875844044096533168406521365747140625
1216777216142606336511705088102288588812709396481039990784579436544223440896596753921084275212800008860027314879671875
1367108864620756992245786214455050240007788560384737955020848477634562251931648745094144174574336283162883025952191748546131869765625
1426843545626843545601162661068828856811520458657628164952005017637632409600205709639688178708480236767641649393228872345600706174441264010923208145546875
Technical details
Node id
n-caf373255c4ddbb9 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-74aba9cdd78cf583

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:      4,       1;
  2:     16,      16,       3;
  3:     64,     112,      44,       5;
  4:    256,     640,     448,     120,      11;
  5:   1024,    3328,    3456,    1504,     292,      21;
  6:   4096,   16384,   22784,   14336,    4528,     704,     43;
  7:  16384,   77824,  136192,  114432,   51392,   12688,   1628,    85;
  8:  65536,  360448,  761856,  808960,  480256,  167040,  33888,  3720,  171;
  9: 262144, 1638400, 4063232, 5242880, 3917824, 1788416, 506880, 87168, 8340, 341;
%F Formula
The row polynomials satisfy 4*n*P(n,x) = (16*n+(24*n-20)*x+(9*n-9)*x^2)*P(n-1,x) - (16*x+40*x^2+9*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[((16*n + (24*n - 20)*x + (9*n - 9)*x^2)*P[n - 1, x] - (16*x + 40*x^2 + 9*x^3)*D[P[n - 1, x], x])/(4*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] = ((16*n + (24*n - 20)*x + (9*n - 9)*x^2)*v[i - 1] - (16*x + 40*x^2 + 9*x^3)*deriv(v[i - 1], x))/(4*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