Real-rooted atlas

A156365 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 4\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-1 - 3\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 13 + 16\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01316433
175134688285
25411268984272163081
346831358614248638410723240005
44729316356021644413523239216422464606033
554583521919343505020281896811783842349121670412810491885
67087261324103966078539259898448329070409796672139187266304256204343641
7102247563524566250112908108813183518089001472963599058567930112082165762639365124422082005

All 15 rows

Technical details
Node id
n-984f75403d58b29b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4358062586543316

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 2)
  • 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:          13,           16,            4;
  1:          75,          134,           68,            8;
  2:         541,         1268,          984,          272,           16;
  3:        4683,        13586,        14248,         6384,         1072,           32;
  4:       47293,       163560,       216444,       135232,        39216,         4224,           64;
  5:      545835,      2191934,      3505020,      2818968,      1178384,       234912,        16704,         128;
  6:     7087261,     32410396,     60785392,     59898448,     32907040,      9796672,      1391872,       66304,        256;
  7:   102247563,    524566250,   1129081088,   1318351808,    900147296,    359905856,     79301120,     8216576,     263936,       512;
  8:  1622632573,   9228981584,  22423353396,  30284929920,  24758721696,  12485395968,   3779445888,   632297472,   48503040,   1052672,    1024;
  9: 28091567595, 175430841494, 474921849652, 728574493224, 694477837536, 423828596160, 164295898752, 38671029504, 5000402688, 286797312, 4203520, 2048;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+5+(2*n+4)*x)*P(n-1,x) - (1+3*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 13 + 16*x + 4*x^2.
%t Mathematica
P[0, x_] := 13 + 16*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(2*n + 5 + (2*n + 4)*x)*P[n - 1, x] - (1 + 3*x + 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 13 + 16*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 5 + (2*n + 4)*x)*v[i - 1] - (1 + 3*x + 2*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 - 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