Real-rooted atlas

A169803 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(40 + 192\,x + 128\,x^{2} + 40\,n + 272\,n x + 320\,n x^{2} + 80\,n^{2} x + 256\,n^{2} x^{2} + 64\,n^{3} x^{2}\right)\,P_{n}(x) = \left(40 + 192\,x + 128\,x^{2} + 120\,n + 1216\,n x + 3648\,n x^{2} + 3072\,n x^{3} + 320\,n^{2} x + 1920\,n^{2} x^{2} + 2560\,n^{2} x^{3} + 192\,n^{3} x^{2} + 512\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 420\,x - 2528\,x^{2} - 6528\,x^{3} - 6144\,x^{4} - 100\,n x - 1120\,n x^{2} - 4160\,n x^{3} - 5120\,n x^{4} - 80\,n^{2} x^{2} - 576\,n^{2} x^{3} - 1024\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
231619
35404893
4880144232
513160432256861
621304105610242405
7345682448358412807914
8551040532810240640023063
9891880111842688025600614471777
10144336022752655368320036864211856
1123359524521615206424320016588828672641225
1237710464881283389446540806021122007041894809
13610182801690567321601657600193536010035201310725647658
14987317603199681541120400640056770564014080104857616639947
Technical details
Node id
n-cded572742f81719 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-83336882d0e5b15c

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:  2,    8;
  2:  3,   16;
  3:  5,   40,    48;
  4:  8,   80,   144;
  5: 13,  160,   432,   256;
  6: 21,  304,  1056,  1024;
  7: 34,  568,  2448,  3584,  1280;
  8: 55, 1040,  5328, 10240,  6400;
  9: 89, 1880, 11184, 26880, 25600, 6144;
%F Formula
The row polynomials satisfy (40*n+40+(80*n^2+272*n+192)*x+(64*n^3+256*n^2+320*n+128)*x^2)*P(n,x) = (120*n+40+(320*n^2+1216*n+192)*x+(192*n^3+1920*n^2+3648*n+128)*x^2+(512*n^3+2560*n^2+3072*n)*x^3)*P(n-1,x) - (25+(100*n+420)*x+(80*n^2+1120*n+2528)*x^2+(576*n^2+4160*n+6528)*x^3+(1024*n^2+5120*n+6144)*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[((120*n + 40 + (320*n^2 + 1216*n + 192)*x + (192*n^3 + 1920*n^2 + 3648*n + 128)*x^2 + (512*n^3 + 2560*n^2 + 3072*n)*x^3)*P[n - 1, x] - (25 + (100*n + 420)*x + (80*n^2 + 1120*n + 2528)*x^2 + (576*n^2 + 4160*n + 6528)*x^3 + (1024*n^2 + 5120*n + 6144)*x^4)*D[P[n - 1, x], x])/(40*n + 40 + (80*n^2 + 272*n + 192)*x + (64*n^3 + 256*n^2 + 320*n + 128)*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] = ((120*n + 40 + (320*n^2 + 1216*n + 192)*x + (192*n^3 + 1920*n^2 + 3648*n + 128)*x^2 + (512*n^3 + 2560*n^2 + 3072*n)*x^3)*v[i - 1] - (25 + (100*n + 420)*x + (80*n^2 + 1120*n + 2528)*x^2 + (576*n^2 + 4160*n + 6528)*x^3 + (1024*n^2 + 5120*n + 6144)*x^4)*deriv(v[i - 1], x))/(40*n + 40 + (80*n^2 + 272*n + 192)*x + (64*n^3 + 256*n^2 + 320*n + 128)*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.