Real-rooted atlas

A062196 → D → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
282432
31512090225
4243607202401344
535840315028005257350
648168010080168008400100838016
7633024264607056066150211681764189189

All 15 rows

Technical details
Node id
n-5b2cf56653b8394b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5f98bd4aea48ed7d

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  3;
  2:  8,   24;
  3: 15,  120,     90;
  4: 24,  360,    720,    240;
  5: 35,  840,   3150,   2800,     525;
  6: 48, 1680,  10080,  16800,    8400,    1008;
  7: 63, 3024,  26460,  70560,   66150,   21168,   1764;
  8: 80, 5040,  60480, 235200,  352800,  211680,  47040,  2880;
  9: 99, 7920, 124740, 665280, 1455300, 1397088, 582120, 95040, 4455;
%F Formula
The row polynomials satisfy (n^4-2*n^2+1)*P(n,x) = (n^4+2*n^3-n^2-2*n+(3*n^4+4*n^3-3*n^2+2*n)*x)*P(n-1,x) + ((2*n^3+4*n^2)*x-(2*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 + 2*n^3 - n^2 - 2*n + (3*n^4 + 4*n^3 - 3*n^2 + 2*n)*x)*P[n - 1, x] + ((2*n^3 + 4*n^2)*x - (2*n^3 + 4*n^2)*x^2)*D[P[n - 1, x], x])/(n^4 - 2*n^2 + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((n^4 + 2*n^3 - n^2 - 2*n + (3*n^4 + 4*n^3 - 3*n^2 + 2*n)*x)*v[i - 1] + ((2*n^3 + 4*n^2)*x - (2*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^4 - 2*n^2 + 1)); 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. [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.