Real-rooted atlas

A128099 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-44\,n - 1008\,n x + 512\,n x^{2} - 240\,n^{2} + 512\,n^{2} x + 128\,n^{3}\right)\,P_{n}(x) = \left(624 + 4480\,x + 5888\,x^{2} - 8192\,x^{3} + 164\,n - 3696\,n x - 13568\,n x^{2} + 8192\,n x^{3} - 1584\,n^{2} - 3328\,n^{2} x + 8192\,n^{2} x^{2} + 640\,n^{3} + 2048\,n^{3} x\right)\,P_{n-1}(x) + \left(63 + 440\,x - 528\,x^{2} - 9216\,x^{3} - 16384\,x^{4} - 36\,n - 1424\,n x - 9216\,n x^{2} - 16384\,n x^{3} - 288\,n^{2} - 2176\,n^{2} x - 4096\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
211819
3211637
46718464315
5149512192853
6371186423045125051
783751048320204816309
81923144243398425600409680027
942773734410886411264020480283605
10949195592342912509440266240327681256443
1120773235984993024185139213721601966084669941
1245219573560279072064122886799360265420826214419537499
139768513669767530944204185602746777615597568183500874314517
142099713215752198251526230272010350592083951616256901122097152300798395
Technical details
Node id
n-fb3a599e2d6a6d19 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-11fdbba636f90041

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:   11,     8;
  3:   21,    16;
  4:   67,   184,     64;
  5:  149,   512,    192;
  6:  371,  1864,   2304,    512;
  7:  837,  5104,   8320,   2048;
  8: 1923, 14424,  33984,  25600,  4096;
  9: 4277, 37344, 108864, 112640, 20480;
%F Formula
The row polynomials satisfy (128*n^3-240*n^2-44*n+(512*n^2-1008*n)*x+512*n*x^2)*P(n,x) = (640*n^3-1584*n^2+164*n+624+(2048*n^3-3328*n^2-3696*n+4480)*x+(8192*n^2-13568*n+5888)*x^2+(8192*n-8192)*x^3)*P(n-1,x) - (288*n^2+36*n-63+(2176*n^2+1424*n-440)*x+(4096*n^2+9216*n+528)*x^2+(16384*n+9216)*x^3+16384*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[((640*n^3 - 1584*n^2 + 164*n + 624 + (2048*n^3 - 3328*n^2 - 3696*n + 4480)*x + (8192*n^2 - 13568*n + 5888)*x^2 + (8192*n - 8192)*x^3)*P[n - 1, x] - (288*n^2 + 36*n - 63 + (2176*n^2 + 1424*n - 440)*x + (4096*n^2 + 9216*n + 528)*x^2 + (16384*n + 9216)*x^3 + 16384*x^4)*D[P[n - 1, x], x])/(128*n^3 - 240*n^2 - 44*n + (512*n^2 - 1008*n)*x + 512*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] = ((640*n^3 - 1584*n^2 + 164*n + 624 + (2048*n^3 - 3328*n^2 - 3696*n + 4480)*x + (8192*n^2 - 13568*n + 5888)*x^2 + (8192*n - 8192)*x^3)*v[i - 1] - (288*n^2 + 36*n - 63 + (2176*n^2 + 1424*n - 440)*x + (4096*n^2 + 9216*n + 528)*x^2 + (16384*n + 9216)*x^3 + 16384*x^4)*deriv(v[i - 1], x))/(128*n^3 - 240*n^2 - 44*n + (512*n^2 - 1008*n)*x + 512*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

References

  1. [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
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.