Real-rooted atlas

A008288 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-1 + 4\,x + 40\,x^{2} + 48\,x^{3} + n - 7\,n x - 36\,n x^{2} - 40\,n x^{3} + 3\,n^{2} x + 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x + 23\,x^{2} + 68\,x^{3} + 48\,x^{4} - n x - 10\,n x^{2} - 24\,n x^{3} - 16\,n x^{4}\right)\,P_{n-1}'(x) + \left(x^{2} + 9\,x^{3} + 16\,x^{4} + 8\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
315511
418181239
5111405829139
61147116417570495
71171113576015071691763
81201606641550204814284086279

All 15 rows

Technical details
Node id
n-9577f69826f076c7 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f8b12148df73b33e

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: 1,  2;
   3: 1,  5,   5;
   4: 1,  8,  18,   12;
   5: 1, 11,  40,   58,   29;
   6: 1, 14,  71,  164,  175,    70;
   7: 1, 17, 111,  357,  601,   507,   169;
   8: 1, 20, 160,  664, 1550,  2048,  1428,   408;
   9: 1, 23, 218, 1112, 3346,  6106,  6632,  3940,   985;
  10: 1, 26, 285, 1728, 6394, 15012, 22442, 20680, 10701, 2378;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n-1+(3*n^2-7*n+4)*x+(8*n^2-36*n+40)*x^2+(8*n^2-40*n+48)*x^3)*P(n-1,x) - ((n-1)*x+(10*n-23)*x^2+(24*n-68)*x^3+(16*n-48)*x^4)*d/dx P(n-1,x) + (x^2+9*x^3+16*x^4+8*x^5)*d^2/dx^2 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[((n - 1 + (3*n^2 - 7*n + 4)*x + (8*n^2 - 36*n + 40)*x^2 + (8*n^2 - 40*n + 48)*x^3)*P[n - 1, x] - ((n - 1)*x + (10*n - 23)*x^2 + (24*n - 68)*x^3 + (16*n - 48)*x^4)*D[P[n - 1, x], x] + (x^2 + 9*x^3 + 16*x^4 + 8*x^5)*D[P[n - 1, x], {x, 2}])/(n - 1)]];
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] = ((n - 1 + (3*n^2 - 7*n + 4)*x + (8*n^2 - 36*n + 40)*x^2 + (8*n^2 - 40*n + 48)*x^3)*v[i - 1] - ((n - 1)*x + (10*n - 23)*x^2 + (24*n - 68)*x^3 + (16*n - 48)*x^4)*deriv(v[i - 1], x) + (x^2 + 9*x^3 + 16*x^4 + 8*x^5)*deriv(deriv(v[i - 1], x), x))/(n - 1)); 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

Candidate equalities ?

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