Real-rooted atlas

A052553 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-10 - 2\,x + 10\,n - 3\,n x + n x^{2} + 5\,n^{2} x - 2\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-50 - 60\,x - 16\,x^{2} + 30\,n - 4\,n x - 27\,n x^{2} - 4\,n x^{3} + 20\,n^{2} x + 12\,n^{2} x^{2} - 2\,n^{2} x^{3} + 3\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 55\,x - 38\,x^{2} - 8\,x^{3} - 25\,n x - 50\,n x^{2} - 29\,n x^{3} - 4\,n x^{4} - 5\,n^{2} x^{2} - 9\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
3224
4347
5510318
6820937
7134027484
821766616179
934142153565390
105526033316025833
118947069942010061784
1214484014221024325363791
1323314882826237695016278042
1437726165508529625555884916989
156104570105661144064751890245835804
Technical details
Node id
n-602e5f94214b4713 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-88dd3d68aa6e666a

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:  1;
   3:  2,   2;
   4:  3,   4;
   5:  5,  10,   3;
   6:  8,  20,   9;
   7: 13,  40,  27,   4;
   8: 21,  76,  66,  16;
   9: 34, 142, 153,  56,  5;
  10: 55, 260, 333, 160, 25;
%F Formula
The row polynomials satisfy (10*n-10+(5*n^2-3*n-2)*x+(n^3-2*n^2+n)*x^2)*P(n,x) = (30*n-50+(20*n^2-4*n-60)*x+(3*n^3+12*n^2-27*n-16)*x^2+(2*n^3-2*n^2-4*n)*x^3)*P(n-1,x) - (25+(25*n+55)*x+(5*n^2+50*n+38)*x^2+(9*n^2+29*n+8)*x^3+(4*n^2+4*n)*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[((30*n - 50 + (20*n^2 - 4*n - 60)*x + (3*n^3 + 12*n^2 - 27*n - 16)*x^2 + (2*n^3 - 2*n^2 - 4*n)*x^3)*P[n - 1, x] - (25 + (25*n + 55)*x + (5*n^2 + 50*n + 38)*x^2 + (9*n^2 + 29*n + 8)*x^3 + (4*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(10*n - 10 + (5*n^2 - 3*n - 2)*x + (n^3 - 2*n^2 + 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] = ((30*n - 50 + (20*n^2 - 4*n - 60)*x + (3*n^3 + 12*n^2 - 27*n - 16)*x^2 + (2*n^3 - 2*n^2 - 4*n)*x^3)*v[i - 1] - (25 + (25*n + 55)*x + (5*n^2 + 50*n + 38)*x^2 + (9*n^2 + 29*n + 8)*x^3 + (4*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(10*n - 10 + (5*n^2 - 3*n - 2)*x + (n^3 - 2*n^2 + 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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.