Real-rooted atlas

A102662 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + x\right)\,P_{n-2}(x) + \left(2 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
297117
3222810161
4539456131217
512829124493161773
63098579294991391912753
77462442325022648861942219805
8180167961072892064674143225825134921
943481858133944346542170486182164331281124373
1010497501071039931231869194245046146363109413311442961
1125342133640310618418963363428212356851002334942945043411577629
1261181353226908976137572313609479251284422381495433545957466043715618809
131477049266152615436439000348800083786987210706084977324810451749749271340120011685
143565892415341741968913680141168866891474005793684734398387153098339280773083955983143171272673
Technical details
Node id
n-e2c7ec533520b86d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f73fb03abea1aab7

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:    9,     7,     1;
  3:   22,    28,    10,     1;
  4:   53,    94,    56,    13,     1;
  5:  128,   291,   244,    93,    16,    1;
  6:  309,   857,   929,   499,   139,   19,    1;
  7:  746,  2442,  3250,  2264,   886,  194,   22,   1;
  8: 1801,  6796, 10728,  9206,  4674, 1432,  258,  25,  1;
  9: 4348, 18581, 33944, 34654, 21704, 8618, 2164, 331, 28, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + (2+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x] + (2 + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 2] + (2 + x)*v[i - 1]); 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