Real-rooted atlas

A026729 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(14 + 68\,x + 8\,x^{2} - 16\,x^{3} - 2\,n - 100\,n x - 24\,n x^{2} + 16\,n x^{3} - 20\,n^{2} + 32\,n^{2} x + 16\,n^{2} x^{2} + 8\,n^{3}\right)\,P_{n}(x) = \left(30 - 248\,x - 64\,x^{2} + 96\,x^{3} + 32\,x^{4} - 184\,n + 144\,n x + 128\,n x^{2} + 112\,n^{2} - 16\,n^{2} x - 32\,n^{2} x^{2} - 16\,n^{3}\right)\,P_{n-1}(x) + \left(5 - 38\,x - 56\,x^{2} + 144\,x^{3} + 144\,x^{4} + 32\,x^{5} - 30\,n + 8\,n x + 208\,n x^{2} + 160\,n x^{3} + 32\,n x^{4} + 20\,n^{2} + 48\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
3814426
41754448123
53515823212016561
668404868808304322484
71299482696370424967366410773
82392098744413624135847136172812845981

All 15 rows

Technical details
Node id
n-f09a1f04142f8fd5 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-304b7b37ebb5392c

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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:   3,    2;
   3:   8,   14,     4;
   4:  17,   54,    44,      8;
   5:  35,  158,   232,    120,     16;
   6:  68,  404,   868,    808,    304,     32;
   7: 129,  948,  2696,   3704,   2496,    736,     64;
   8: 239, 2098,  7444,  13624,  13584,   7136,   1728,   128;
   9: 436, 4450, 18944,  43464,  57936,  45024,  19328,  3968,  256;
  10: 785, 9138, 45412, 125480, 210336, 218880, 138816, 50304, 8960, 512;
%F Formula
The row polynomials satisfy (8*n^3-20*n^2-2*n+14+(32*n^2-100*n+68)*x+(16*n^2-24*n+8)*x^2+(16*n-16)*x^3)*P(n,x) = -(16*n^3-112*n^2+184*n-30+(16*n^2-144*n+248)*x+(32*n^2-128*n+64)*x^2-96*x^3-32*x^4)*P(n-1,x) + (20*n^2-30*n+5+(48*n^2+8*n-38)*x+(16*n^2+208*n-56)*x^2+(160*n+144)*x^3+(32*n+144)*x^4+32*x^5)*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[(-(16*n^3 - 112*n^2 + 184*n - 30 + (16*n^2 - 144*n + 248)*x + (32*n^2 - 128*n + 64)*x^2 - 96*x^3 - 32*x^4)*P[n - 1, x] + (20*n^2 - 30*n + 5 + (48*n^2 + 8*n - 38)*x + (16*n^2 + 208*n - 56)*x^2 + (160*n + 144)*x^3 + (32*n + 144)*x^4 + 32*x^5)*D[P[n - 1, x], x])/(8*n^3 - 20*n^2 - 2*n + 14 + (32*n^2 - 100*n + 68)*x + (16*n^2 - 24*n + 8)*x^2 + (16*n - 16)*x^3)]];
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] = (-(16*n^3 - 112*n^2 + 184*n - 30 + (16*n^2 - 144*n + 248)*x + (32*n^2 - 128*n + 64)*x^2 - 96*x^3 - 32*x^4)*v[i - 1] + (20*n^2 - 30*n + 5 + (48*n^2 + 8*n - 38)*x + (16*n^2 + 208*n - 56)*x^2 + (160*n + 144)*x^3 + (32*n + 144)*x^4 + 32*x^5)*deriv(v[i - 1], x))/(8*n^3 - 20*n^2 - 2*n + 14 + (32*n^2 - 100*n + 68)*x + (16*n^2 - 24*n + 8)*x^2 + (16*n - 16)*x^3)); 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

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.