Real-rooted atlas

A113214 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n}(x) = \left(20 + 92\,x + 156\,x^{2} + 116\,x^{3} + 32\,x^{4} - 11\,n - 56\,n x - 111\,n x^{2} - 98\,n x^{3} - 32\,n x^{4}\right)\,P_{n-1}(x) + \left(5 + 43\,x + 131\,x^{2} + 185\,x^{3} + 124\,x^{4} + 32\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
24232855
3760162136365
41113156110166562375
518270157544466048316815525
629528391515091318123470415296101375
747100490244415212675421314419414473856662125
876186519711117026426289974288136278410657283566084324375
912334084140328915812780033702966702681684140165765376172185628243125
10199614584375677811352489812388815297780604829808050557824308293128313856184459375
11322109641678901525116912554737738020109986978225770280319692864297265408163301888401428481204728125
12521193923276423320942224774641069647573667607859124618321633934448205284761617170552328582369281938268167868234375
138433405062920870399325318391928634083211297990793306162222717227395211387396256128568460809772171776448069632093587865651388453125
141364594191192094145934841217363897317123533266656833109994833232806020958853948859248769234622087885445964854922615808232607457284518821888335624609375
Technical details
Node id
n-33d2e4cdfcb2bfa1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9ad9a93745a082c8

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,    6;
  2:   4,   23,    28;
  3:   7,   60,   162,    136;
  4:  11,  131,   561,   1016,     656;
  5:  18,  270,  1575,   4446,    6048,    3168;
  6:  29,  528,  3915,  15091,   31812,   34704,   15296;
  7:  47, 1004,  9024,  44152,  126754,  213144,  194144,   73856;
  8:  76, 1865, 19711, 117026,  426289,  974288, 1362784, 1065728,  356608;
  9: 123, 3408, 41403, 289158, 1278003, 3702966, 7026816, 8414016, 5765376, 1721856;
%F Formula
The row polynomials satisfy (3*n+6*n*x+3*n*x^2)*P(n,x) = -(11*n-20+(56*n-92)*x+(111*n-156)*x^2+(98*n-116)*x^3+(32*n-32)*x^4)*P(n-1,x) + (5+43*x+131*x^2+185*x^3+124*x^4+32*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[(-(11*n - 20 + (56*n - 92)*x + (111*n - 156)*x^2 + (98*n - 116)*x^3 + (32*n - 32)*x^4)*P[n - 1, x] + (5 + 43*x + 131*x^2 + 185*x^3 + 124*x^4 + 32*x^5)*D[P[n - 1, x], x])/(3*n + 6*n*x + 3*n*x^2)]];
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] = (-(11*n - 20 + (56*n - 92)*x + (111*n - 156)*x^2 + (98*n - 116)*x^3 + (32*n - 32)*x^4)*v[i - 1] + (5 + 43*x + 131*x^2 + 185*x^3 + 124*x^4 + 32*x^5)*deriv(v[i - 1], x))/(3*n + 6*n*x + 3*n*x^2)); 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