Real-rooted atlas

A257614 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 5\,n\right)\,P_{n}(x) = \left(12 - 96\,x - 48\,x^{2} + 30\,n - 180\,n x - 10\,n x^{2} + 150\,n^{2} x + 225\,n^{2} x^{2} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 30\,x^{3} + 75\,n x - 50\,n x^{2} - 25\,n x^{3} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-12 + 18\,x - 5\,n - 60\,n x + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-15\,x + 15\,x^{2} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
24561272
38488732321260
41636881756873768028728
5322652030486040648066300192804384
664187368453348013820320755580056210444826659584
7128131567261990164367927520459909400123980328460485210241019729088

All 15 rows

Technical details
Node id
n-58613cbd6cc60a77 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-598bdb4e63e884bb

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,        4;
  2:   4,       56,          12;
  3:   8,      488,         732,           32;
  4:  16,     3688,       17568,         7376,           80;
  5:  32,    26520,      304860,       406480,        66300,          192;
  6:  64,   187368,     4533480,     13820320,      7555800,       562104,          448;
  7: 128,  1315672,    61990164,    367927520,    459909400,    123980328,      4604852,        1024;
  8: 256,  9219176,   807034224,   8486413744,  20236013600,  12729620616,   1883079856,    36876704,      2304;
  9: 512, 64555736, 10196074956, 178702493872, 731608763060, 877930515672, 312729364276, 27189533216, 290500812, 5120;
%F Formula
The row polynomials satisfy (5*n-3)*P(n,x) = (30*n+12+(150*n^2-180*n-96)*x-(125*n^3-225*n^2+10*n+48)*x^2)*P(n-2,x) + ((75*n+30)*x-(125*n^2+50*n)*x^2+(125*n^2-25*n-30)*x^3)*d/dx P(n-2,x) - (5*n+12-(50*n^2-60*n+18)*x)*P(n-1,x) + ((25*n-15)*x-(25*n-15)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((30*n + 12 + (150*n^2 - 180*n - 96)*x - (125*n^3 - 225*n^2 + 10*n + 48)*x^2)*P[n - 2, x] + ((75*n + 30)*x - (125*n^2 + 50*n)*x^2 + (125*n^2 - 25*n - 30)*x^3)*D[P[n - 2, x], x] - (5*n + 12 - (50*n^2 - 60*n + 18)*x)*P[n - 1, x] + ((25*n - 15)*x - (25*n - 15)*x^2)*D[P[n - 1, x], x])/(5*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((30*n + 12 + (150*n^2 - 180*n - 96)*x - (125*n^3 - 225*n^2 + 10*n + 48)*x^2)*v[i - 2] + ((75*n + 30)*x - (125*n^2 + 50*n)*x^2 + (125*n^2 - 25*n - 30)*x^3)*deriv(v[i - 2], x) - (5*n + 12 - (50*n^2 - 60*n + 18)*x)*v[i - 1] + ((25*n - 15)*x - (25*n - 15)*x^2)*deriv(v[i - 1], x))/(5*n - 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.