Real-rooted atlas

A090452 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(2 - 2\,x^{2} + 4\,n + 9\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 4\,x^{2} - 2\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12316
2515167144
3146311410551121360
4422526449107713961181813152
513299032706240758060833235111023025128896
64293861156423762059625653315046427483103652600402331273856
714301501572072209209409332569415578690434493240751971882782553906514212661504

All 15 rows

Technical details
Node id
n-fb5f5aabac06a450 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-2df49debe483a37b

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:     2,      3,       1;
  2:     5,     15,      16,       7,        1;
  3:    14,     63,     114,     105,       51,       12,        1;
  4:    42,    252,     644,     910,      771,      396,      118,       18,        1;
  5:   132,    990,    3270,    6240,     7580,     6083,     3235,     1110,      230,       25,        1;
  6:   429,   3861,   15642,   37620,    59625,    65331,    50464,    27483,    10365,     2600,      402,      33,       1;
  7:  1430,  15015,   72072,  209209,   409332,   569415,   578690,   434493,   240751,    97188,    27825,    5390,     651,     42,      1;
  8:  4862,  58344,  323752, 1101100,  2564562,  4328324,  5462072,  5241060,  3849715,  2161288,   918484,  289884,   65828,  10220,    996,    52,    1;
  9: 16796, 226746, 1428102, 5569200, 15049008, 29875482, 45072846, 52723476, 48344922, 34895685, 19793907, 8757108, 2979312, 762048, 141552, 18096, 1458, 63, 1;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (4*n+2+9*n*x+(5*n-2)*x^2)*P(n-1,x) - (2*x+4*x^2+2*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 2 + 9*n*x + (5*n - 2)*x^2)*P[n - 1, x] - (2*x + 4*x^2 + 2*x^3)*D[P[n - 1, x], x])/(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; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 2 + 9*n*x + (5*n - 2)*x^2)*v[i - 1] - (2*x + 4*x^2 + 2*x^3)*deriv(v[i - 1], x))/(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

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