Real-rooted atlas

A128099 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(72\,n + 208\,n x + 144\,n^{2} x + 128\,n^{2} x^{2} + 128\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-288 - 2560\,x - 6656\,x^{2} - 4096\,x^{3} + 360\,n + 1904\,n x + 1536\,n x^{2} - 2048\,n x^{3} + 1008\,n^{2} x + 4736\,n^{2} x^{2} + 4096\,n^{2} x^{3} + 640\,n^{3} x^{2} + 2048\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-81 - 1080\,x - 5264\,x^{2} - 11008\,x^{3} - 8192\,x^{4} - 324\,n x - 3312\,n x^{2} - 11136\,n x^{3} - 12288\,n x^{4} - 288\,n^{2} x^{2} - 2176\,n^{2} x^{3} - 4096\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
231619
353237
411112192315
521256576853
643656230420485051
78515046528819216309

All 15 rows

Technical details
Node id
n-e543b64520110d38 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6699deb0f226dac0

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 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:   1;
  2:   3,   16;
  3:   5,   32;
  4:  11,  112,   192;
  5:  21,  256,   576;
  6:  43,  656,  2304,   2048;
  7:  85, 1504,  6528,   8192;
  8: 171, 3504, 19008,  36864,  20480;
  9: 341, 7872, 50112, 122880, 102400;
%F Formula
The row polynomials satisfy (72*n+(144*n^2+208*n)*x+(128*n^3+128*n^2)*x^2)*P(n,x) = (360*n-288+(1008*n^2+1904*n-2560)*x+(640*n^3+4736*n^2+1536*n-6656)*x^2+(2048*n^3+4096*n^2-2048*n-4096)*x^3)*P(n-1,x) - (81+(324*n+1080)*x+(288*n^2+3312*n+5264)*x^2+(2176*n^2+11136*n+11008)*x^3+(4096*n^2+12288*n+8192)*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[((360*n - 288 + (1008*n^2 + 1904*n - 2560)*x + (640*n^3 + 4736*n^2 + 1536*n - 6656)*x^2 + (2048*n^3 + 4096*n^2 - 2048*n - 4096)*x^3)*P[n - 1, x] - (81 + (324*n + 1080)*x + (288*n^2 + 3312*n + 5264)*x^2 + (2176*n^2 + 11136*n + 11008)*x^3 + (4096*n^2 + 12288*n + 8192)*x^4)*D[P[n - 1, x], x])/(72*n + (144*n^2 + 208*n)*x + (128*n^3 + 128*n^2)*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] = ((360*n - 288 + (1008*n^2 + 1904*n - 2560)*x + (640*n^3 + 4736*n^2 + 1536*n - 6656)*x^2 + (2048*n^3 + 4096*n^2 - 2048*n - 4096)*x^3)*v[i - 1] - (81 + (324*n + 1080)*x + (288*n^2 + 3312*n + 5264)*x^2 + (2176*n^2 + 11136*n + 11008)*x^3 + (4096*n^2 + 12288*n + 8192)*x^4)*deriv(v[i - 1], x))/(72*n + (144*n^2 + 208*n)*x + (128*n^3 + 128*n^2)*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

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.