Real-rooted atlas

type_b_eulerian → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(1 - 6\,x - 3\,x^{2} + 2\,n - 8\,n x + 2\,n x^{2} + 8\,n^{2} x + 12\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x - 2\,x^{3} + 4\,n x - 4\,n x^{2} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 2\,x - 8\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
2112316
3146694120
4115269030451152
51474504667281185613440
61144431629941925271543327184320
714358181971103889212986153639421525382903040

All 15 rows

Technical details
Node id
n-362bfef25c84aa1f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2e37e4aadfbd49d5

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: 1,    12,       3;
  3: 1,    46,      69,        4;
  4: 1,   152,     690,      304,         5;
  5: 1,   474,    5046,     6728,      1185,         6;
  6: 1,  1444,   31629,    94192,     52715,      4332,         7;
  7: 1,  4358,  181971,  1038892,   1298615,    363942,     15253,        8;
  8: 1, 13104,  994836,  9941152,  23375070,  14911728,   2321284,    52416,      9;
  9: 1, 39346, 5269020, 86831888, 347066470, 416479764, 151955804, 14050720, 177057, 10;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+(8*n^2-8*n-6)*x-(8*n^3-12*n^2-2*n+3)*x^2)*P(n-2,x) + ((4*n+2)*x-(8*n^2+4*n)*x^2+(8*n^2-2)*x^3)*d/dx P(n-2,x) - (2-(8*n^2-8*n+2)*x)*P(n-1,x) + ((4*n-2)*x-(4*n-2)*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[Together[((2*n + 1 + (8*n^2 - 8*n - 6)*x - (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*P[n - 2, x] + ((4*n + 2)*x - (8*n^2 + 4*n)*x^2 + (8*n^2 - 2)*x^3)*D[P[n - 2, x], x] - (2 - (8*n^2 - 8*n + 2)*x)*P[n - 1, x] + ((4*n - 2)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = ((2*n + 1 + (8*n^2 - 8*n - 6)*x - (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*v[i - 2] + ((4*n + 2)*x - (8*n^2 + 4*n)*x^2 + (8*n^2 - 2)*x^3)*deriv(v[i - 2], x) - (2 - (8*n^2 - 8*n + 2)*x)*v[i - 1] + ((4*n - 2)*x - (4*n - 2)*x^2)*deriv(v[i - 1], x))/(2*n - 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.