Real-rooted atlas

A128099 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-34\,n - 92\,n x + 64\,n x^{2} - 12\,n^{2} + 64\,n^{2} x + 16\,n^{3}\right)\,P_{n}(x) = \left(120 + 352\,x - 32\,x^{2} - 512\,x^{3} - 122\,n - 604\,n x - 416\,n x^{2} + 512\,n x^{3} - 108\,n^{2} + 96\,n^{2} x + 512\,n^{2} x^{2} + 80\,n^{3} + 128\,n^{3} x\right)\,P_{n-1}(x) + \left(-9 - 196\,x - 932\,x^{2} - 1664\,x^{3} - 1024\,x^{4} - 90\,n - 628\,n x - 1408\,n x^{2} - 1024\,n x^{3} - 72\,n^{2} - 272\,n^{2} x - 256\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
27411
313821
4396016115
58516048293
6207548384641203
7461146413122563493
8104740445040217625612563
9230910320156968960128038565
105087261004828838080115201024130099
111106963848137664134144550406144407909
12239751540123820324533762560005836840961331859
135157336480010210401419776999936315392286724201189
1411044785370026666884275520367360015912962867201638413474355
Technical details
Node id
n-87a5e8350ee7b970 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2b56890a9e86bb73

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:    7,     4;
  3:   13,     8;
  4:   39,    60,    16;
  5:   85,   160,    48;
  6:  207,   548,   384,   64;
  7:  461,  1464,  1312,  256;
  8: 1047,  4044,  5040, 2176,  256;
  9: 2309, 10320, 15696, 8960, 1280;
%F Formula
The row polynomials satisfy (16*n^3-12*n^2-34*n+(64*n^2-92*n)*x+64*n*x^2)*P(n,x) = (80*n^3-108*n^2-122*n+120+(128*n^3+96*n^2-604*n+352)*x+(512*n^2-416*n-32)*x^2+(512*n-512)*x^3)*P(n-1,x) - (72*n^2+90*n+9+(272*n^2+628*n+196)*x+(256*n^2+1408*n+932)*x^2+(1024*n+1664)*x^3+1024*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[((80*n^3 - 108*n^2 - 122*n + 120 + (128*n^3 + 96*n^2 - 604*n + 352)*x + (512*n^2 - 416*n - 32)*x^2 + (512*n - 512)*x^3)*P[n - 1, x] - (72*n^2 + 90*n + 9 + (272*n^2 + 628*n + 196)*x + (256*n^2 + 1408*n + 932)*x^2 + (1024*n + 1664)*x^3 + 1024*x^4)*D[P[n - 1, x], x])/(16*n^3 - 12*n^2 - 34*n + (64*n^2 - 92*n)*x + 64*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] = ((80*n^3 - 108*n^2 - 122*n + 120 + (128*n^3 + 96*n^2 - 604*n + 352)*x + (512*n^2 - 416*n - 32)*x^2 + (512*n - 512)*x^3)*v[i - 1] - (72*n^2 + 90*n + 9 + (272*n^2 + 628*n + 196)*x + (256*n^2 + 1408*n + 932)*x^2 + (1024*n + 1664)*x^3 + 1024*x^4)*deriv(v[i - 1], x))/(16*n^3 - 12*n^2 - 34*n + (64*n^2 - 92*n)*x + 64*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.