Real-rooted atlas

A099040 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n + 16\,n x + 8\,n x^{2} + 6\,n^{2} x + 10\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(24 + 100\,x + 136\,x^{2} + 64\,x^{3} - 12\,n + 10\,n x + 96\,n x^{2} + 80\,n x^{3} - 18\,n^{2} x - 12\,n^{2} x^{2} + 16\,n^{2} x^{3} - 4\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(9 + 54\,x + 116\,x^{2} + 104\,x^{3} + 32\,x^{4} + 18\,n x + 78\,n x^{2} + 104\,n x^{3} + 40\,n x^{4} + 6\,n^{2} x^{2} + 16\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
26404894
316160384256816
4445762064281612806780
5120195293121920017920614454648
632863683792010470414848010444828672430920
78962022414438449971295232010199045734401310723341952

All 15 rows

Technical details
Node id
n-d3a973da6773d88c · 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 0)
  • 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:    6,     40,      48;
  3:   16,    160,     384,     256;
  4:   44,    576,    2064,    2816,     1280;
  5:  120,   1952,    9312,   19200,    17920,     6144;
  6:  328,   6368,   37920,  104704,   148480,   104448,    28672;
  7:  896,  20224,  144384,  499712,   952320,  1019904,   573440,   131072;
  8: 2448,  62976,  524160, 2181120,  5223680,  7532544,  6451200,  3014656,   589824;
  9: 6688, 193152, 1836288, 8927232, 25756160, 46749696, 53960704, 38404096, 15335424, 2621440;
%F Formula
The row polynomials satisfy (6*n+(6*n^2+16*n)*x+(2*n^3+10*n^2+8*n)*x^2)*P(n,x) = -(12*n-24+(18*n^2-10*n-100)*x+(4*n^3+12*n^2-96*n-136)*x^2-(16*n^2+80*n+64)*x^3)*P(n-1,x) + (9+(18*n+54)*x+(6*n^2+78*n+116)*x^2+(16*n^2+104*n+104)*x^3+(8*n^2+40*n+32)*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[(-(12*n - 24 + (18*n^2 - 10*n - 100)*x + (4*n^3 + 12*n^2 - 96*n - 136)*x^2 - (16*n^2 + 80*n + 64)*x^3)*P[n - 1, x] + (9 + (18*n + 54)*x + (6*n^2 + 78*n + 116)*x^2 + (16*n^2 + 104*n + 104)*x^3 + (8*n^2 + 40*n + 32)*x^4)*D[P[n - 1, x], x])/(6*n + (6*n^2 + 16*n)*x + (2*n^3 + 10*n^2 + 8*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] = (-(12*n - 24 + (18*n^2 - 10*n - 100)*x + (4*n^3 + 12*n^2 - 96*n - 136)*x^2 - (16*n^2 + 80*n + 64)*x^3)*v[i - 1] + (9 + (18*n + 54)*x + (6*n^2 + 78*n + 116)*x^2 + (16*n^2 + 104*n + 104)*x^3 + (8*n^2 + 40*n + 32)*x^4)*deriv(v[i - 1], x))/(6*n + (6*n^2 + 16*n)*x + (2*n^3 + 10*n^2 + 8*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.