Real-rooted atlas

A130850 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
226311
362421455
424120150605359
5120720117072015562891
672050401008084003010378727635
75040403209576010080051030115928898305439

All 15 rows

Technical details
Node id
n-3e467fc9dff41715 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d5a52fbc92ca7963

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 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:      1,       2;
  2:      2,       6,        3;
  3:      6,      24,       21,        4;
  4:     24,     120,      150,       60,        5;
  5:    120,     720,     1170,      720,      155,       6;
  6:    720,    5040,    10080,     8400,     3010,     378,       7;
  7:   5040,   40320,    95760,   100800,    51030,   11592,     889,      8;
  8:  40320,  362880,   997920,  1270080,   834120,  279720,   42350,   2040,    9;
  9: 362880, 3628800, 11340000, 16934400, 13696200, 6123600, 1432410, 149280, 4599, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-1+(2*n^2-2)*x+(n^2-1)*x^2)*P(n-2,x) + ((n+1)*x+(2*n+2)*x^2+(n+1)*x^3)*d/dx P(n-2,x) + (2*n+1+2*n*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 1 + (2*n^2 - 2)*x + (n^2 - 1)*x^2)*P[n - 2, x] + ((n + 1)*x + (2*n + 2)*x^2 + (n + 1)*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 2*n*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 1 + (2*n^2 - 2)*x + (n^2 - 1)*x^2)*v[i - 2] + ((n + 1)*x + (2*n + 2)*x^2 + (n + 1)*x^3)*deriv(v[i - 2], x) + (2*n + 1 + 2*n*x)*v[i - 1] - (x + x^2)*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

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. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.