Real-rooted atlas

A030528 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(52 - 464\,x - 576\,x^{2} + 256\,x^{3} - 188\,n - 592\,n x - 320\,n x^{2} + 256\,n x^{3} - 176\,n^{2} - 128\,n^{2} x + 256\,n^{2} x^{2} + 64\,n^{3}\right)\,P_{n}(x) = \left(216 - 5056\,x - 7680\,x^{2} - 1024\,x^{3} + 2048\,x^{4} - 2824\,n - 4320\,n x - 384\,n x^{2} + 1536\,n x^{3} + 1248\,n^{2} + 1280\,n^{2} x - 512\,n^{2} x^{2} - 128\,n^{3}\right)\,P_{n-1}(x) + \left(25 - 460\,x - 2784\,x^{2} - 1920\,x^{3} + 1280\,x^{4} + 1024\,x^{5} - 300\,n - 1600\,n x - 1408\,n x^{2} + 1024\,n x^{3} + 1024\,n x^{4} + 80\,n^{2} + 384\,n^{2} x + 256\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
219481683
34523228864629
496760187215362564520
519320687856125447680102431365
63735056261606425675520368644096212325
77021151675696253824456960423936172032163841411050

All 15 rows

Technical details
Node id
n-7c37fc732e912755 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8bbee996c5ae4546

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:    6,     4;
  2:   19,    48,     16;
  3:   45,   232,    288,      64;
  4:   96,   760,   1872,    1536,     256;
  5:  193,  2068,   7856,   12544,    7680,     1024;
  6:  373,  5056,  26160,   64256,   75520,    36864,     4096;
  7:  702, 11516,  75696,  253824,  456960,   423936,   172032,    16384;
  8: 1295, 24952, 199248,  850944, 2091520,  2959360,  2265088,   786432,   65536;
  9: 2353, 52088, 489936, 2548096, 7986432, 15412224, 17920000, 11665408, 3538944, 262144;
%F Formula
The row polynomials satisfy (64*n^3-176*n^2-188*n+52-(128*n^2+592*n+464)*x+(256*n^2-320*n-576)*x^2+(256*n+256)*x^3)*P(n,x) = -(128*n^3-1248*n^2+2824*n-216-(1280*n^2-4320*n-5056)*x+(512*n^2+384*n+7680)*x^2-(1536*n-1024)*x^3-2048*x^4)*P(n-1,x) + (80*n^2-300*n+25+(384*n^2-1600*n-460)*x+(256*n^2-1408*n-2784)*x^2+(1024*n-1920)*x^3+(1024*n+1280)*x^4+1024*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[(-(128*n^3 - 1248*n^2 + 2824*n - 216 - (1280*n^2 - 4320*n - 5056)*x + (512*n^2 + 384*n + 7680)*x^2 - (1536*n - 1024)*x^3 - 2048*x^4)*P[n - 1, x] + (80*n^2 - 300*n + 25 + (384*n^2 - 1600*n - 460)*x + (256*n^2 - 1408*n - 2784)*x^2 + (1024*n - 1920)*x^3 + (1024*n + 1280)*x^4 + 1024*x^5)*D[P[n - 1, x], x])/(64*n^3 - 176*n^2 - 188*n + 52 - (128*n^2 + 592*n + 464)*x + (256*n^2 - 320*n - 576)*x^2 + (256*n + 256)*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] = (-(128*n^3 - 1248*n^2 + 2824*n - 216 - (1280*n^2 - 4320*n - 5056)*x + (512*n^2 + 384*n + 7680)*x^2 - (1536*n - 1024)*x^3 - 2048*x^4)*v[i - 1] + (80*n^2 - 300*n + 25 + (384*n^2 - 1600*n - 460)*x + (256*n^2 - 1408*n - 2784)*x^2 + (1024*n - 1920)*x^3 + (1024*n + 1280)*x^4 + 1024*x^5)*deriv(v[i - 1], x))/(64*n^3 - 176*n^2 - 188*n + 52 - (128*n^2 + 592*n + 464)*x + (256*n^2 - 320*n - 576)*x^2 + (256*n + 256)*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.