Real-rooted atlas

A052553 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-148 + 752\,x - 256\,x^{2} + 388\,n - 1008\,n x + 256\,n x^{2} - 304\,n^{2} + 256\,n^{2} x + 64\,n^{3}\right)\,P_{n}(x) = \left(-420 + 1136\,x + 6656\,x^{2} - 4096\,x^{3} + 1356\,n + 1616\,n x - 7296\,n x^{2} + 2048\,n x^{3} - 976\,n^{2} - 2176\,n^{2} x + 2048\,n^{2} x^{2} + 192\,n^{3} + 512\,n^{3} x\right)\,P_{n-1}(x) + \left(-45 - 24\,x + 1264\,x^{2} + 1536\,x^{3} - 4096\,x^{4} + 140\,n + 688\,n x - 512\,n x^{2} - 4096\,n x^{3} - 80\,n^{2} - 576\,n^{2} x - 1024\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
111
211
36410
411819
525521693
64813648232
79336833664861
817385611202562405

All 15 rows

Technical details
Node id
n-cef261beff6d66d5 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9e1a96db7a3abf55

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   1;
   3:   6,    4;
   4:  11,    8;
   5:  25,   52,   16;
   6:  48,  136,   48;
   7:  93,  368,  336,   64;
   8: 173,  856, 1120,  256;
   9: 318, 1916, 3504, 1920,  256;
  10: 575, 4072, 9456, 7680, 1280;
%F Formula
The row polynomials satisfy (64*n^3-304*n^2+388*n-148+(256*n^2-1008*n+752)*x+(256*n-256)*x^2)*P(n,x) = (192*n^3-976*n^2+1356*n-420+(512*n^3-2176*n^2+1616*n+1136)*x+(2048*n^2-7296*n+6656)*x^2+(2048*n-4096)*x^3)*P(n-1,x) - (80*n^2-140*n+45+(576*n^2-688*n+24)*x+(1024*n^2+512*n-1264)*x^2+(4096*n-1536)*x^3+4096*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((192*n^3 - 976*n^2 + 1356*n - 420 + (512*n^3 - 2176*n^2 + 1616*n + 1136)*x + (2048*n^2 - 7296*n + 6656)*x^2 + (2048*n - 4096)*x^3)*P[n - 1, x] - (80*n^2 - 140*n + 45 + (576*n^2 - 688*n + 24)*x + (1024*n^2 + 512*n - 1264)*x^2 + (4096*n - 1536)*x^3 + 4096*x^4)*D[P[n - 1, x], x])/(64*n^3 - 304*n^2 + 388*n - 148 + (256*n^2 - 1008*n + 752)*x + (256*n - 256)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((192*n^3 - 976*n^2 + 1356*n - 420 + (512*n^3 - 2176*n^2 + 1616*n + 1136)*x + (2048*n^2 - 7296*n + 6656)*x^2 + (2048*n - 4096)*x^3)*v[i - 1] - (80*n^2 - 140*n + 45 + (576*n^2 - 688*n + 24)*x + (1024*n^2 + 512*n - 1264)*x^2 + (4096*n - 1536)*x^3 + 4096*x^4)*deriv(v[i - 1], x))/(64*n^3 - 304*n^2 + 388*n - 148 + (256*n^2 - 1008*n + 752)*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, v[i])))};
T(10)

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.