Real-rooted atlas

A112007 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2616325
324116664210
412088898420852205
57207416132005808570627720
65040679681744201284802805014407405405
74032068227223559122576080979000119700345886756750

All 15 rows

Technical details
Node id
n-1ddcde253f71775a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e6b802c8b5974669

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:       2,        2;
  2:       6,       16,         3;
  3:      24,      116,        66,         4;
  4:     120,      888,       984,       208,         5;
  5:     720,     7416,     13200,      5808,       570,         6;
  6:    5040,    67968,    174420,    128480,     28050,      1440,        7;
  7:   40320,   682272,   2355912,   2576080,    979000,    119700,     3458,       8;
  8:  362880,  7467840,  33078888,  49760256,  28827500,   6375000,   470820,    8032,     9;
  9: 3628800, 88678080, 486560736, 955619616, 776786920, 268590000, 37283120, 1750784, 18234, 10;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3-n+(2*n^3-2*n^2-2*n+2)*x+(n^3-n)*x^2)*P(n-2,x) - ((n^2-1)*x+(n+1)*x^2-(n^2+n)*x^3)*d/dx P(n-2,x) + (2*n^2+n-1+2*n^2*x)*P(n-1,x) + (n*x-n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n + (2*n^3 - 2*n^2 - 2*n + 2)*x + (n^3 - n)*x^2)*P[n - 2, x] - ((n^2 - 1)*x + (n + 1)*x^2 - (n^2 + n)*x^3)*D[P[n - 2, x], x] + (2*n^2 + n - 1 + 2*n^2*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];
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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - n + (2*n^3 - 2*n^2 - 2*n + 2)*x + (n^3 - n)*x^2)*v[i - 2] - ((n^2 - 1)*x + (n + 1)*x^2 - (n^2 + n)*x^3)*deriv(v[i - 2], x) + (2*n^2 + n - 1 + 2*n^2*x)*v[i - 1] + (n*x - n*x^2)*deriv(v[i - 1], x))/(n)); 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.