Real-rooted atlas

A052553 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-40 - 32\,x + 40\,n - 48\,n x + 64\,n x^{2} + 80\,n^{2} x - 128\,n^{2} x^{2} + 64\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-200 - 960\,x - 1024\,x^{2} + 120\,n - 64\,n x - 1728\,n x^{2} - 1024\,n x^{3} + 320\,n^{2} x + 768\,n^{2} x^{2} - 512\,n^{2} x^{3} + 192\,n^{3} x^{2} + 512\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 220\,x - 608\,x^{2} - 512\,x^{3} - 100\,n x - 800\,n x^{2} - 1856\,n x^{3} - 1024\,n x^{4} - 80\,n^{2} x^{2} - 576\,n^{2} x^{3} - 1024\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
32810
431619
55404893
6880144232
713160432256861
821304105610242405
9345682448358412807914
10551040532810240640023063
11891880111842688025600614471777
12144336022752655368320036864211856
1323359524521615206424320016588828672641225
1437710464881283389446540806021122007041894809
15610182801690567321601657600193536010035201310725647658
Technical details
Node id
n-06b69d03f9ece2e8 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-83336882d0e5b15c

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,    8;
   4:  3,   16;
   5:  5,   40,   48;
   6:  8,   80,  144;
   7: 13,  160,  432,   256;
   8: 21,  304, 1056,  1024;
   9: 34,  568, 2448,  3584, 1280;
  10: 55, 1040, 5328, 10240, 6400;
%F Formula
The row polynomials satisfy (40*n-40+(80*n^2-48*n-32)*x+(64*n^3-128*n^2+64*n)*x^2)*P(n,x) = (120*n-200+(320*n^2-64*n-960)*x+(192*n^3+768*n^2-1728*n-1024)*x^2+(512*n^3-512*n^2-1024*n)*x^3)*P(n-1,x) - (25+(100*n+220)*x+(80*n^2+800*n+608)*x^2+(576*n^2+1856*n+512)*x^3+(1024*n^2+1024*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[((120*n - 200 + (320*n^2 - 64*n - 960)*x + (192*n^3 + 768*n^2 - 1728*n - 1024)*x^2 + (512*n^3 - 512*n^2 - 1024*n)*x^3)*P[n - 1, x] - (25 + (100*n + 220)*x + (80*n^2 + 800*n + 608)*x^2 + (576*n^2 + 1856*n + 512)*x^3 + (1024*n^2 + 1024*n)*x^4)*D[P[n - 1, x], x])/(40*n - 40 + (80*n^2 - 48*n - 32)*x + (64*n^3 - 128*n^2 + 64*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] = ((120*n - 200 + (320*n^2 - 64*n - 960)*x + (192*n^3 + 768*n^2 - 1728*n - 1024)*x^2 + (512*n^3 - 512*n^2 - 1024*n)*x^3)*v[i - 1] - (25 + (100*n + 220)*x + (80*n^2 + 800*n + 608)*x^2 + (576*n^2 + 1856*n + 512)*x^3 + (1024*n^2 + 1024*n)*x^4)*deriv(v[i - 1], x))/(40*n - 40 + (80*n^2 - 48*n - 32)*x + (64*n^3 - 128*n^2 + 64*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.