Real-rooted atlas

A100861 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 64\,x + 32\,x^{2} - 16\,n + 32\,n x + 8\,n^{2}\right)\,P_{n}(x) = \left(-8 - 8\,x + 128\,x^{2} - 128\,x^{3} + 10\,n + 8\,n x - 224\,n x^{2} + 128\,n x^{3} - 16\,n^{2} - 64\,n^{2} x + 128\,n^{2} x^{2} + 8\,n^{3} + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(-1 - 8\,x - 32\,x^{2} - 128\,x^{3} - 256\,x^{4} - 16\,n x - 128\,n x^{2} - 256\,n x^{3} - 4\,n^{2} - 32\,n^{2} x - 64\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
26410
3161228
45814448250
51866402401066
6676348043209609436
724161562426880672051640

All 15 rows

Technical details
Node id
n-62d4fc6a50b318be · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-29515cc327d9f4a5

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:     6,      4;
  3:    16,     12;
  4:    58,    144,      48;
  5:   186,    640,     240;
  6:   676,   3480,    4320,     960;
  7:  2416,  15624,   26880,    6720;
  8:  9276,  75712,  194880,  161280,  26880;
  9: 36028, 347904, 1124928, 1290240, 241920;
%F Formula
The row polynomials satisfy (8*n^2-16*n+2+(32*n-64)*x+32*x^2)*P(n,x) = (8*n^3-16*n^2+10*n-8+(32*n^3-64*n^2+8*n-8)*x+(128*n^2-224*n+128)*x^2+(128*n-128)*x^3)*P(n-1,x) - (4*n^2+1+(32*n^2+16*n+8)*x+(64*n^2+128*n+32)*x^2+(256*n+128)*x^3+256*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[((8*n^3 - 16*n^2 + 10*n - 8 + (32*n^3 - 64*n^2 + 8*n - 8)*x + (128*n^2 - 224*n + 128)*x^2 + (128*n - 128)*x^3)*P[n - 1, x] - (4*n^2 + 1 + (32*n^2 + 16*n + 8)*x + (64*n^2 + 128*n + 32)*x^2 + (256*n + 128)*x^3 + 256*x^4)*D[P[n - 1, x], x])/(8*n^2 - 16*n + 2 + (32*n - 64)*x + 32*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] = ((8*n^3 - 16*n^2 + 10*n - 8 + (32*n^3 - 64*n^2 + 8*n - 8)*x + (128*n^2 - 224*n + 128)*x^2 + (128*n - 128)*x^3)*v[i - 1] - (4*n^2 + 1 + (32*n^2 + 16*n + 8)*x + (64*n^2 + 128*n + 32)*x^2 + (256*n + 128)*x^3 + 256*x^4)*deriv(v[i - 1], x))/(8*n^2 - 16*n + 2 + (32*n - 64)*x + 32*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.