Real-rooted atlas

A088617 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n + n x - n x^{2} - 3\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(-2\,x + 2\,x^{3} + 2\,n + 5\,n x - n x^{2} - 2\,n x^{3} - n^{2} - 4\,n^{2} x + n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(4\,x + 2\,x^{2} - 4\,x^{3} - 2\,x^{4} - 4\,n x - 4\,n x^{2} + 2\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
347213
472625563
511701359114321
616155490644336421683
72230114002940294012541328989
8295323402102901617013068471942948639

All 15 rows

Technical details
Node id
n-cc0179e01d57c43d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-695598672c6e8bd2

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:  2,    1;
   3:  4,    7,     2;
   4:  7,   26,    25,     5;
   5: 11,   70,   135,    91,     14;
   6: 16,  155,   490,   644,    336,     42;
   7: 22,  301,  1400,  2940,   2940,   1254,    132;
   8: 29,  532,  3402, 10290,  16170,  13068,   4719,    429;
   9: 37,  876,  7350, 30030,  66990,  84084,  57057,  17875,  1430;
  10: 46, 1365, 14520, 76692, 228228, 402402, 420420, 245960, 68068, 4862;
%F Formula
The row polynomials satisfy (n^3-3*n^2+4*n+n*x-n*x^2)*P(n,x) = (n^3-n^2+2*n+(2*n^3-4*n^2+5*n-2)*x-n*x^2-(2*n-2)*x^3)*P(n-1,x) + ((2*n^2-4*n+4)*x+(2*n^2-4*n+2)*x^2-4*x^3-2*x^4)*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[Together[((n^3 - n^2 + 2*n + (2*n^3 - 4*n^2 + 5*n - 2)*x - n*x^2 - (2*n - 2)*x^3)*P[n - 1, x] + ((2*n^2 - 4*n + 4)*x + (2*n^2 - 4*n + 2)*x^2 - 4*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 4*n + n*x - n*x^2)]];
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] = ((n^3 - n^2 + 2*n + (2*n^3 - 4*n^2 + 5*n - 2)*x - n*x^2 - (2*n - 2)*x^3)*v[i - 1] + ((2*n^2 - 4*n + 4)*x + (2*n^2 - 4*n + 2)*x^2 - 4*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 4*n + n*x - n*x^2)); 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. [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.