Real-rooted atlas

A169803 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-76 - 240\,x + 256\,x^{2} - 60\,n + 16\,n x + 256\,n x^{2} + 80\,n^{2} + 256\,n^{2} x + 64\,n^{3}\right)\,P_{n}(x) = \left(-76 - 240\,x + 256\,x^{2} - 244\,n - 944\,n x + 896\,n x^{2} + 2048\,n x^{3} + 176\,n^{2} + 896\,n^{2} x + 2048\,n^{2} x^{2} + 192\,n^{3} + 512\,n^{3} x\right)\,P_{n-1}(x) + \left(-85 - 952\,x - 3856\,x^{2} - 6656\,x^{3} - 4096\,x^{4} - 180\,n - 1616\,n x - 4608\,n x^{2} - 4096\,n x^{3} - 80\,n^{2} - 576\,n^{2} x - 1024\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4Pn(1)
011
16410
211819
325521693
44813648232
59336833664861
617385611202562405
73181916350419202567914

All 15 rows

Technical details
Node id
n-20b0b45c124254ce · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9e1a96db7a3abf55

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:    6,    4;
  2:   11,    8;
  3:   25,   52,    16;
  4:   48,  136,    48;
  5:   93,  368,   336,    64;
  6:  173,  856,  1120,   256;
  7:  318, 1916,  3504,  1920,   256;
  8:  575, 4072,  9456,  7680,  1280;
  9: 1029, 8396, 23888, 27200, 10240, 1024;
%F Formula
The row polynomials satisfy (64*n^3+80*n^2-60*n-76+(256*n^2+16*n-240)*x+(256*n+256)*x^2)*P(n,x) = (192*n^3+176*n^2-244*n-76+(512*n^3+896*n^2-944*n-240)*x+(2048*n^2+896*n+256)*x^2+2048*n*x^3)*P(n-1,x) - (80*n^2+180*n+85+(576*n^2+1616*n+952)*x+(1024*n^2+4608*n+3856)*x^2+(4096*n+6656)*x^3+4096*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[((192*n^3 + 176*n^2 - 244*n - 76 + (512*n^3 + 896*n^2 - 944*n - 240)*x + (2048*n^2 + 896*n + 256)*x^2 + 2048*n*x^3)*P[n - 1, x] - (80*n^2 + 180*n + 85 + (576*n^2 + 1616*n + 952)*x + (1024*n^2 + 4608*n + 3856)*x^2 + (4096*n + 6656)*x^3 + 4096*x^4)*D[P[n - 1, x], x])/(64*n^3 + 80*n^2 - 60*n - 76 + (256*n^2 + 16*n - 240)*x + (256*n + 256)*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] = ((192*n^3 + 176*n^2 - 244*n - 76 + (512*n^3 + 896*n^2 - 944*n - 240)*x + (2048*n^2 + 896*n + 256)*x^2 + 2048*n*x^3)*v[i - 1] - (80*n^2 + 180*n + 85 + (576*n^2 + 1616*n + 952)*x + (1024*n^2 + 4608*n + 3856)*x^2 + (4096*n + 6656)*x^3 + 4096*x^4)*deriv(v[i - 1], x))/(64*n^3 + 80*n^2 - 60*n - 76 + (256*n^2 + 16*n - 240)*x + (256*n + 256)*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. [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.