Real-rooted atlas

A062145 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + n + n^{2}\right)\,P_{n}(x) = \left(2 + 2\,x + 3\,n + n x + 7\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-2 - 3\,x - x^{2} - 4\,n - 6\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
29413
3615010121
4377438165201000
5224133321806420357834
613073235601652056009105659719
7755171594621366206048014490176484448417
8433905104881010593455770801820703301231501203337492

All 15 rows

Technical details
Node id
n-ff62bf19b729c3e5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6042166219f3d43f

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 -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:        1;
   2:        9,        4;
   3:       61,       50,       10;
   4:      377,      438,      165,       20;
   5:     2241,     3332,     1806,      420,       35;
   6:    13073,    23560,    16520,     5600,      910,      56;
   7:    75517,   159462,   136620,    60480,    14490,    1764,      84;
   8:   433905,  1048810,  1059345,   577080,   182070,   33012,    3150,    120;
   9:  2485825,  6761480,  7855540,  5072760,  1981980,  476784,   68376,   5280,  165;
  10: 14218905, 42955020, 56378520, 42060480, 19577250, 5865552, 1123584, 131472, 8415, 220;
%F Formula
The row polynomials satisfy (n^2+n-2)*P(n,x) = (7*n^2+3*n+2+(3*n^2+n+2)*x)*P(n-1,x) - (4*n+2+(6*n+3)*x+(2*n+1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((7*n^2 + 3*n + 2 + (3*n^2 + n + 2)*x)*P[n - 1, x] - (4*n + 2 + (6*n + 3)*x + (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 + n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((7*n^2 + 3*n + 2 + (3*n^2 + n + 2)*x)*v[i - 1] - (4*n + 2 + (6*n + 3)*x + (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 + n - 2)); 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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.