Real-rooted atlas

A099093 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-22\,n - 76\,n x - 40\,n x^{2} + 48\,n x^{3} - 4\,n^{2} + 32\,n^{2} x + 48\,n^{2} x^{2} + 8\,n^{3}\right)\,P_{n}(x) = \left(-146 - 488\,x - 320\,x^{2} + 288\,x^{3} + 288\,x^{4} - 128\,n - 96\,n x + 224\,n x^{2} + 192\,n x^{3} + 128\,n^{2} + 80\,n^{2} x - 96\,n^{2} x^{2} - 16\,n^{3}\right)\,P_{n-1}(x) + \left(-21 - 46\,x + 200\,x^{2} + 752\,x^{3} + 816\,x^{4} + 288\,x^{5} - 14\,n + 72\,n x + 464\,n x^{2} + 672\,n x^{3} + 288\,n x^{4} + 28\,n^{2} + 80\,n^{2} x + 48\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19615
25411436204
3279102610442162565
413417074126728424129630807
5615642444109836128952635047776358668
62740523360479315213493521174176458784466564083129
71193131212246510202811413224144102249945504321926427993645701739

All 15 rows

Technical details
Node id
n-6da0a27d960dd213 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-73b6f4e40d633899

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 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:       9,        6;
  2:      54,      114,        36;
  3:     279,     1026,      1044,       216;
  4:    1341,     7074,     12672,      8424,       1296;
  5:    6156,    42444,    109836,    128952,      63504,       7776;
  6:   27405,   233604,    793152,   1349352,    1174176,     458784,      46656;
  7:  119313,  1212246,   5102028,  11413224,   14410224,    9945504,    3219264,    279936;
  8:  510786,  6027534,  30293352,  84138264,  139852656,  140115744,   80061696,  22114944,   1679616;
  9: 2158083, 29012742, 169587756, 563163192, 1163390688, 1535417856, 1273662144, 620618112, 149485824, 10077696;
%F Formula
The row polynomials satisfy (8*n^3-4*n^2-22*n+(32*n^2-76*n)*x+(48*n^2-40*n)*x^2+48*n*x^3)*P(n,x) = -(16*n^3-128*n^2+128*n+146-(80*n^2-96*n-488)*x+(96*n^2-224*n+320)*x^2-(192*n+288)*x^3-288*x^4)*P(n-1,x) + (28*n^2-14*n-21+(80*n^2+72*n-46)*x+(48*n^2+464*n+200)*x^2+(672*n+752)*x^3+(288*n+816)*x^4+288*x^5)*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[(-(16*n^3 - 128*n^2 + 128*n + 146 - (80*n^2 - 96*n - 488)*x + (96*n^2 - 224*n + 320)*x^2 - (192*n + 288)*x^3 - 288*x^4)*P[n - 1, x] + (28*n^2 - 14*n - 21 + (80*n^2 + 72*n - 46)*x + (48*n^2 + 464*n + 200)*x^2 + (672*n + 752)*x^3 + (288*n + 816)*x^4 + 288*x^5)*D[P[n - 1, x], x])/(8*n^3 - 4*n^2 - 22*n + (32*n^2 - 76*n)*x + (48*n^2 - 40*n)*x^2 + 48*n*x^3)]];
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] = (-(16*n^3 - 128*n^2 + 128*n + 146 - (80*n^2 - 96*n - 488)*x + (96*n^2 - 224*n + 320)*x^2 - (192*n + 288)*x^3 - 288*x^4)*v[i - 1] + (28*n^2 - 14*n - 21 + (80*n^2 + 72*n - 46)*x + (48*n^2 + 464*n + 200)*x^2 + (672*n + 752)*x^3 + (288*n + 816)*x^4 + 288*x^5)*deriv(v[i - 1], x))/(8*n^3 - 4*n^2 - 22*n + (32*n^2 - 76*n)*x + (48*n^2 - 40*n)*x^2 + 48*n*x^3)); 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.