Real-rooted atlas

reversed rows → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + 4\,n x^{2} + 2\,n^{2} + 8\,n^{2} x + 12\,n^{2} x^{2} + n^{3} + 9\,n^{3} x + 13\,n^{3} x^{2} + 3\,n^{4} x + 6\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(n + 10\,n x + 48\,n x^{2} + 72\,n x^{3} + 2\,n^{2} + 24\,n^{2} x + 86\,n^{2} x^{2} + 120\,n^{2} x^{3} + n^{3} + 17\,n^{3} x + 61\,n^{3} x^{2} + 74\,n^{3} x^{3} + 3\,n^{4} x + 16\,n^{4} x^{2} + 20\,n^{4} x^{3} + n^{5} x^{2} + 2\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-x - 18\,x^{2} - 92\,x^{3} - 144\,x^{4} - 2\,n x - 32\,n x^{2} - 156\,n x^{3} - 240\,n x^{4} - n^{2} x - 19\,n^{2} x^{2} - 97\,n^{2} x^{3} - 148\,n^{2} x^{4} - 3\,n^{3} x^{2} - 22\,n^{3} x^{3} - 40\,n^{3} x^{4} - n^{4} x^{3} - 4\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1189
211213
31161835
41204566
51248132138
6128126112267
713218025650519
8136243480225985
9140315800625721853
10144396123213753963444
11148486179226251296986346
1215258524964550327663711597
13156693336073507056240112821045
1416081044001125013608686096037949
Technical details
Node id
n-7d770cc3e0cbe6ef · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c5ae5c76bbcc6daf

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  8;
  2: 1, 12;
  3: 1, 16,  18;
  4: 1, 20,  45;
  5: 1, 24,  81,  32;
  6: 1, 28, 126, 112;
  7: 1, 32, 180, 256,  50;
  8: 1, 36, 243, 480, 225;
  9: 1, 40, 315, 800, 625, 72;
%F Formula
The row polynomials satisfy (n^3+2*n^2+n+(3*n^4+9*n^3+8*n^2+2*n)*x+(n^5+6*n^4+13*n^3+12*n^2+4*n)*x^2)*P(n,x) = (n^3+2*n^2+n+(3*n^4+17*n^3+24*n^2+10*n)*x+(n^5+16*n^4+61*n^3+86*n^2+48*n)*x^2+(2*n^5+20*n^4+74*n^3+120*n^2+72*n)*x^3)*P(n-1,x) - ((n^2+2*n+1)*x+(3*n^3+19*n^2+32*n+18)*x^2+(n^4+22*n^3+97*n^2+156*n+92)*x^3+(4*n^4+40*n^3+148*n^2+240*n+144)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 2*n^2 + n + (3*n^4 + 17*n^3 + 24*n^2 + 10*n)*x + (n^5 + 16*n^4 + 61*n^3 + 86*n^2 + 48*n)*x^2 + (2*n^5 + 20*n^4 + 74*n^3 + 120*n^2 + 72*n)*x^3)*P[n - 1, x] - ((n^2 + 2*n + 1)*x + (3*n^3 + 19*n^2 + 32*n + 18)*x^2 + (n^4 + 22*n^3 + 97*n^2 + 156*n + 92)*x^3 + (4*n^4 + 40*n^3 + 148*n^2 + 240*n + 144)*x^4)*D[P[n - 1, x], x])/(n^3 + 2*n^2 + n + (3*n^4 + 9*n^3 + 8*n^2 + 2*n)*x + (n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^3 + 2*n^2 + n + (3*n^4 + 17*n^3 + 24*n^2 + 10*n)*x + (n^5 + 16*n^4 + 61*n^3 + 86*n^2 + 48*n)*x^2 + (2*n^5 + 20*n^4 + 74*n^3 + 120*n^2 + 72*n)*x^3)*v[i - 1] - ((n^2 + 2*n + 1)*x + (3*n^3 + 19*n^2 + 32*n + 18)*x^2 + (n^4 + 22*n^3 + 97*n^2 + 156*n + 92)*x^3 + (4*n^4 + 40*n^3 + 148*n^2 + 240*n + 144)*x^4)*deriv(v[i - 1], x))/(n^3 + 2*n^2 + n + (3*n^4 + 9*n^3 + 8*n^2 + 2*n)*x + (n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n)*x^2)); 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. [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.