Real-rooted atlas

A099040 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12\,n + 64\,n x + 64\,n x^{2} + 24\,n^{2} x + 80\,n^{2} x^{2} + 16\,n^{3} x^{2}\right)\,P_{n}(x) = \left(48 + 400\,x + 1088\,x^{2} + 1024\,x^{3} - 24\,n + 40\,n x + 768\,n x^{2} + 1280\,n x^{3} - 72\,n^{2} x - 96\,n^{2} x^{2} + 256\,n^{2} x^{3} - 32\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(9 + 108\,x + 464\,x^{2} + 832\,x^{3} + 512\,x^{4} + 36\,n x + 312\,n x^{2} + 832\,n x^{3} + 640\,n x^{4} + 24\,n^{2} x^{2} + 128\,n^{2} x^{3} + 128\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
121618
2680192278
316320153620483920
44411528256225282048052460
5120390437248153600286720196608678200
6328127361516808376322375680334233618350088555400
789640448577536399769615237120326369283670016016777216105968000

All 15 rows

Technical details
Node id
n-a7171def0ab9d5f0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-21ac0866256495b4

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:    2,     16;
  2:    6,     80,     192;
  3:   16,    320,    1536,     2048;
  4:   44,   1152,    8256,    22528,     20480;
  5:  120,   3904,   37248,   153600,    286720,     196608;
  6:  328,  12736,  151680,   837632,   2375680,    3342336,    1835008;
  7:  896,  40448,  577536,  3997696,  15237120,   32636928,   36700160,   16777216;
  8: 2448, 125952, 2096640, 17448960,  83578880,  241041408,  412876800,  385875968,  150994944;
  9: 6688, 386304, 7345152, 71417856, 412098560, 1495990272, 3453485056, 4915724288, 3925868544, 1342177280;
%F Formula
The row polynomials satisfy (12*n+(24*n^2+64*n)*x+(16*n^3+80*n^2+64*n)*x^2)*P(n,x) = -(24*n-48+(72*n^2-40*n-400)*x+(32*n^3+96*n^2-768*n-1088)*x^2-(256*n^2+1280*n+1024)*x^3)*P(n-1,x) + (9+(36*n+108)*x+(24*n^2+312*n+464)*x^2+(128*n^2+832*n+832)*x^3+(128*n^2+640*n+512)*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[(-(24*n - 48 + (72*n^2 - 40*n - 400)*x + (32*n^3 + 96*n^2 - 768*n - 1088)*x^2 - (256*n^2 + 1280*n + 1024)*x^3)*P[n - 1, x] + (9 + (36*n + 108)*x + (24*n^2 + 312*n + 464)*x^2 + (128*n^2 + 832*n + 832)*x^3 + (128*n^2 + 640*n + 512)*x^4)*D[P[n - 1, x], x])/(12*n + (24*n^2 + 64*n)*x + (16*n^3 + 80*n^2 + 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] = (-(24*n - 48 + (72*n^2 - 40*n - 400)*x + (32*n^3 + 96*n^2 - 768*n - 1088)*x^2 - (256*n^2 + 1280*n + 1024)*x^3)*v[i - 1] + (9 + (36*n + 108)*x + (24*n^2 + 312*n + 464)*x^2 + (128*n^2 + 832*n + 832)*x^3 + (128*n^2 + 640*n + 512)*x^4)*deriv(v[i - 1], x))/(12*n + (24*n^2 + 64*n)*x + (16*n^3 + 80*n^2 + 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.