Real-rooted atlas

A249251 → basis A007318 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-2 - n + 3\,n^{2}\right)\,P_{n-1}(x) + \left(72 + 60\,x + 12\,x^{2} - 33\,n - 24\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(66 + 81\,x + 32\,x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17310
23730673
31992379312541
410691680981252244006
5574311211868833396364829665
63085371970695043555610158153696219673
71657514497215200473321541265132872836001921626706
8890461275490037106552841480135274540989677184825638412045961
9478380716620615255528902281306513031865493893912416401996321862476889201845
10256999579907095017118189017456472011633242552930962166503423575136501408414721536660550798
111380673995847797011121777745128636640097966836052022940319653585352817268989419212303369139230724891461121
12741736909342387516072198828579195698820787830417047826780842109369717680958660159683448265244162962176199680614436221880241
1339848193431991047641945775499796641127703596102862161041690125926210255058627925703123223294021246434384069272832702028843315212288268227545050
142140757053311511150801028656209269243772613188445832095592834795153318819750919868085150814376280164328627000331904130750060817700864016418304933888245761986258456073
Technical details
Node id
n-53d7a4d92b6e4464 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e9530ab7bff93797

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:       7,        3;
  2:      37,       30,        6;
  3:     199,      237,       93,       12;
  4:    1069,     1680,      981,      252,       24;
  5:    5743,    11211,     8688,     3339,      636,      48;
  6:   30853,    71970,    69504,    35556,    10158,    1536,      96;
  7:  165751,   449721,   520047,   332154,   126513,   28728,    3600,    192;
  8:  890461,  2754900,  3710655,  2841480,  1352745,  409896,   77184,   8256,   384;
  9: 4783807, 16620615, 25552890, 22813065, 13031865, 4938939, 1241640, 199632, 18624, 768;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (3*n^2-n-2)*P(n-1,x) - (33*n-72+(24*n-60)*x+(4*n-12)*x^2)*d/dx P(n-1,x) + (66+81*x+32*x^2+4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - n - 2)*P[n - 1, x] - (33*n - 72 + (24*n - 60)*x + (4*n - 12)*x^2)*D[P[n - 1, x], x] + (66 + 81*x + 32*x^2 + 4*x^3)*D[P[n - 1, x], {x, 2}])/(2*n - 2)]];
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] = 7 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n^2 - n - 2)*v[i - 1] - (33*n - 72 + (24*n - 60)*x + (4*n - 12)*x^2)*deriv(v[i - 1], x) + (66 + 81*x + 32*x^2 + 4*x^3)*deriv(deriv(v[i - 1], x), x))/(2*n - 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

Candidate equalities ?

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