Real-rooted atlas

A063967 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-6 - 10\,x + 6\,n + 4\,n x - 6\,n x^{2} + 6\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(36 + 72\,x + 32\,x^{2} - 12\,n + 46\,n x + 108\,n x^{2} + 48\,n x^{3} - 18\,n^{2} x + 16\,n^{2} x^{3} - 4\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(9 + 36\,x + 44\,x^{2} + 16\,x^{3} + 18\,n x + 66\,n x^{2} + 72\,n x^{3} + 24\,n x^{4} + 6\,n^{2} x^{2} + 16\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
22810
36404894
416160384256816
5445762064281612806780
6120195293121920017920614454648
732863683792010470414848010444828672430920
88962022414438449971295232010199045734401310723341952

All 15 rows

Technical details
Node id
n-6a4e81b25e0c9fb0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8baf4a7702ce9f62

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,      8;
   3:    6,     40,      48;
   4:   16,    160,     384,     256;
   5:   44,    576,    2064,    2816,     1280;
   6:  120,   1952,    9312,   19200,    17920,     6144;
   7:  328,   6368,   37920,  104704,   148480,   104448,    28672;
   8:  896,  20224,  144384,  499712,   952320,  1019904,   573440,   131072;
   9: 2448,  62976,  524160, 2181120,  5223680,  7532544,  6451200,  3014656,   589824;
  10: 6688, 193152, 1836288, 8927232, 25756160, 46749696, 53960704, 38404096, 15335424, 2621440;
%F Formula
The row polynomials satisfy (6*n-6+(6*n^2+4*n-10)*x+(2*n^3+4*n^2-6*n)*x^2)*P(n,x) = -(12*n-36+(18*n^2-46*n-72)*x+(4*n^3-108*n-32)*x^2-(16*n^2+48*n)*x^3)*P(n-1,x) + (9+(18*n+36)*x+(6*n^2+66*n+44)*x^2+(16*n^2+72*n+16)*x^3+(8*n^2+24*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[(-(12*n - 36 + (18*n^2 - 46*n - 72)*x + (4*n^3 - 108*n - 32)*x^2 - (16*n^2 + 48*n)*x^3)*P[n - 1, x] + (9 + (18*n + 36)*x + (6*n^2 + 66*n + 44)*x^2 + (16*n^2 + 72*n + 16)*x^3 + (8*n^2 + 24*n)*x^4)*D[P[n - 1, x], x])/(6*n - 6 + (6*n^2 + 4*n - 10)*x + (2*n^3 + 4*n^2 - 6*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] = (-(12*n - 36 + (18*n^2 - 46*n - 72)*x + (4*n^3 - 108*n - 32)*x^2 - (16*n^2 + 48*n)*x^3)*v[i - 1] + (9 + (18*n + 36)*x + (6*n^2 + 66*n + 44)*x^2 + (16*n^2 + 72*n + 16)*x^3 + (8*n^2 + 24*n)*x^4)*deriv(v[i - 1], x))/(6*n - 6 + (6*n^2 + 4*n - 10)*x + (2*n^3 + 4*n^2 - 6*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.