Real-rooted atlas

A304236 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(52\,n + 76\,n x + 52\,n^{2} x + 24\,n^{2} x^{2} + 24\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-312 - 1392\,x - 1824\,x^{2} - 576\,x^{3} + 364\,n + 1000\,n x + 432\,n x^{2} - 288\,n x^{3} + 520\,n^{2} x + 1272\,n^{2} x^{2} + 576\,n^{2} x^{3} + 168\,n^{3} x^{2} + 288\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-169 - 1144\,x - 2836\,x^{2} - 3024\,x^{3} - 1152\,x^{4} - 338\,n x - 1768\,n x^{2} - 3048\,n x^{3} - 1728\,n x^{4} - 156\,n^{2} x^{2} - 600\,n^{2} x^{3} - 576\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
241216
372431
419108108235
540264324628
69781616208643397
721720884968345610729

All 15 rows

Technical details
Node id
n-7685d05c3fb614e8 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-34977177fd3c37a4

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:    1;
  2:    4,    12;
  3:    7,    24;
  4:   19,   108,   108;
  5:   40,   264,   324;
  6:   97,   816,  1620,   864;
  7:  217,  2088,  4968,  3456;
  8:  508,  5700, 17172, 19008,  6480;
  9: 1159, 14568, 50868, 69120, 32400;
%F Formula
The row polynomials satisfy (52*n+(52*n^2+76*n)*x+(24*n^3+24*n^2)*x^2)*P(n,x) = (364*n-312+(520*n^2+1000*n-1392)*x+(168*n^3+1272*n^2+432*n-1824)*x^2+(288*n^3+576*n^2-288*n-576)*x^3)*P(n-1,x) - (169+(338*n+1144)*x+(156*n^2+1768*n+2836)*x^2+(600*n^2+3048*n+3024)*x^3+(576*n^2+1728*n+1152)*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[((364*n - 312 + (520*n^2 + 1000*n - 1392)*x + (168*n^3 + 1272*n^2 + 432*n - 1824)*x^2 + (288*n^3 + 576*n^2 - 288*n - 576)*x^3)*P[n - 1, x] - (169 + (338*n + 1144)*x + (156*n^2 + 1768*n + 2836)*x^2 + (600*n^2 + 3048*n + 3024)*x^3 + (576*n^2 + 1728*n + 1152)*x^4)*D[P[n - 1, x], x])/(52*n + (52*n^2 + 76*n)*x + (24*n^3 + 24*n^2)*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] = ((364*n - 312 + (520*n^2 + 1000*n - 1392)*x + (168*n^3 + 1272*n^2 + 432*n - 1824)*x^2 + (288*n^3 + 576*n^2 - 288*n - 576)*x^3)*v[i - 1] - (169 + (338*n + 1144)*x + (156*n^2 + 1768*n + 2836)*x^2 + (600*n^2 + 3048*n + 3024)*x^3 + (576*n^2 + 1728*n + 1152)*x^4)*deriv(v[i - 1], x))/(52*n + (52*n^2 + 76*n)*x + (24*n^3 + 24*n^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.