Real-rooted atlas

A154602 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
31438
419191140
51167010449240
61251905506412571664
7136425208045314380153912992
8149833626522491395153280310299112256

All 15 rows

Technical details
Node id
n-985f1a76b6ca48a8 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-b3b7fb695f9c1be8

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,  1;
   3: 1,  4,    3;
   4: 1,  9,   19,    11;
   5: 1, 16,   70,   104,     49;
   6: 1, 25,  190,   550,    641,     257;
   7: 1, 36,  425,  2080,   4531,    4380,    1539;
   8: 1, 49,  833,  6265,  22491,   39515,   32803,   10299;
   9: 1, 64, 1484, 16016,  87206,  247072,  365324,  266768,   75905;
  10: 1, 81, 2460, 36204, 281190, 1192086, 2792476, 3575820, 2337505, 609441;
%F Formula
The row polynomials satisfy P(n,x) = (1+(2*n-3)*x+(2*n-4)*x^2)*P(n-1,x) - (2*x^2+2*x^3)*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[(1 + (2*n - 3)*x + (2*n - 4)*x^2)*P[n - 1, x] - (2*x^2 + 2*x^3)*D[P[n - 1, x], x]];
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] = (1 + (2*n - 3)*x + (2*n - 4)*x^2)*v[i - 1] - (2*x^2 + 2*x^3)*deriv(v[i - 1], x)); 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

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.