Real-rooted atlas

A257180 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 3\,x + n + n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 42 + 42\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
14242993
2336504222271089
330246048407410508114277
4302407560068880277204686243207369
53326409979201149120635040171402202027293307053
63991680139708801947456013759200518145699842485182218757463569
75189184020756736033995808029338848014294750439076128558839435385065611080778197
8726485760326918592061551705606286896000377658288013523176802791639203041892014547661968321877696089

All 15 rows

Technical details
Node id
n-e23fac9b1096b64b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-09455fa73ad50cf5

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:           42,           42,             9;
   2:          336,          504,           222,            27;
   3:         3024,         6048,          4074,          1050,            81;
   4:        30240,        75600,         68880,         27720,          4686,           243;
   5:       332640,       997920,       1149120,        635040,        171402,         20202,          729;
   6:      3991680,     13970880,      19474560,      13759200,       5181456,        998424,        85182,         2187;
   7:     51891840,    207567360,     339958080,     293388480,     142947504,      39076128,      5588394,       353850,        6561;
   8:    726485760,   3269185920,    6155170560,    6286896000,    3776582880,    1352317680,    279163920,     30418920,     1454766,     19683;
   9:  10897286400,  54486432000,  115978262400,  136994457600,   98124475680,   43735839840,  11978699040,   1918334880,   162277962,   5937162,    59049;
  10: 174356582400, 958961203200, 2277532857600, 3056688835200, 2547438284160, 1364214519360, 469673709120, 101115907200, 12808232976, 852949944, 24102942, 177147;
%F Formula
The row polynomials satisfy P(n,x) = (n+6+(n+3)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 42 + 42*x + 9*x^2.
%t Mathematica
P[1, x_] := 42 + 42*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 6 + (n + 3)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 42 + 42*x + 9*x^2; for(n = 2, N, my(i = n); v[i] = (n + 6 + (n + 3)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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