Real-rooted atlas

A306364 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4\,n + n^{2} x + n^{3} x\right)\,P_{n}(x) = \left(16\,x - 4\,x^{2} - 8\,n - 8\,n x - 4\,n x^{2} - 4\,n^{2} - 6\,n^{2} x + 3\,n^{2} x^{2} + 3\,n^{3} x + 4\,n^{3} x^{2} + n^{4} x + n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-16\,x + 24\,x^{2} - 8\,x^{3} - 8\,n x + 24\,n x^{2} - 16\,n x^{3} + 10\,n^{2} x^{2} - 10\,n^{2} x^{3} + 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
133
2121224
360180240
436021604052925
52520252001417541895
62016030240034020025200687960
718144038102407144200158760012723480
8181440050803200142884000635040002480625261486225
919958400718502400282910320020956320002455818755908777875
1023950080010777536000565820640006286896000014734912500353637900145556611200
11311351040017124307200011558907360001798052256000702364162500505702197003881233956600
12435891456002876883609600242737054560005034546316800029499294825000424789845480068831687925111355666346925
136538371840005099930035200052593028488000014024807596800001150472498175000276113399562000134221791453753420072258978375
141046139494400095198693990400011780838381312000392694612710400004295097326520000015462350375472000150328406428200017531009496000111946886701650000
Technical details
Node id
n-7872228541dd86d5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-60e5382a5ccf99ce

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:        3;
  2:       12,        12;
  3:       60,       180;
  4:      360,      2160,        405;
  5:     2520,     25200,      14175;
  6:    20160,    302400,     340200,      25200;
  7:   181440,   3810240,    7144200,    1587600;
  8:  1814400,  50803200,  142884000,   63504000,   2480625;
  9: 19958400, 718502400, 2829103200, 2095632000, 245581875;
%F Formula
The row polynomials satisfy -(4*n-(n^3+n^2)*x)*P(n,x) = -(4*n^2+8*n-(n^4+3*n^3-6*n^2-8*n+16)*x-(n^4+4*n^3+3*n^2-4*n-4)*x^2)*P(n-1,x) - ((8*n+16)*x-(2*n^3+10*n^2+24*n+24)*x^2+(2*n^3+10*n^2+16*n+8)*x^3)*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[(-(4*n^2 + 8*n - (n^4 + 3*n^3 - 6*n^2 - 8*n + 16)*x - (n^4 + 4*n^3 + 3*n^2 - 4*n - 4)*x^2)*P[n - 1, x] - ((8*n + 16)*x - (2*n^3 + 10*n^2 + 24*n + 24)*x^2 + (2*n^3 + 10*n^2 + 16*n + 8)*x^3)*D[P[n - 1, x], x])/(-4*n + (n^3 + n^2)*x)]];
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] = (-(4*n^2 + 8*n - (n^4 + 3*n^3 - 6*n^2 - 8*n + 16)*x - (n^4 + 4*n^3 + 3*n^2 - 4*n - 4)*x^2)*v[i - 1] - ((8*n + 16)*x - (2*n^3 + 10*n^2 + 24*n + 24)*x^2 + (2*n^3 + 10*n^2 + 16*n + 8)*x^3)*deriv(v[i - 1], x))/(-4*n + (n^3 + n^2)*x)); 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.