Real-rooted atlas

A308737 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,n x - n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 16\,x - 6\,x^{2} - 8\,x^{3} - 2\,n x + 21\,n x^{2} - n x^{4} - 3\,n^{2} x^{2} + 9\,n^{2} x^{3} - n^{3} x^{3} + n^{3} x^{4}\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2} - 10\,x^{3} - 4\,x^{4} + 3\,n x^{2} - n x^{3} - 5\,n x^{4} - n x^{5} + n^{2} x^{3} - n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1167
21162138
31349360188
4168288376155888
51134774152812653784080
61264193551206375381688918400
71522463515380255752298610717204081856

All 15 rows

Technical details
Node id
n-bac037884d2391c5 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-dae00e5755e6d397

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: 1,    6;
  2: 1,   16,    21;
  3: 1,   34,    93,     60;
  4: 1,   68,   288,    376,    155;
  5: 1,  134,   774,   1528,   1265,    378;
  6: 1,  264,  1935,   5120,   6375,   3816,    889;
  7: 1,  522,  4635,  15380,  25575,  22986,  10717,   2040;
  8: 1, 1036, 10794,  43064,  89600, 107436,  75166,  28624,  4599;
  9: 1, 2062, 24636, 114800, 286790, 429996, 401212, 229216, 73665, 10230;
%F Formula
The row polynomials satisfy (2+2*n*x+(n^2-n)*x^2)*P(n,x) = (2-(2*n-16)*x-(3*n^2-21*n+6)*x^2-(n^3-9*n^2+8)*x^3+(n^3-n)*x^4)*P(n-1,x) + (2*x+(3*n-4)*x^2+(n^2-n-10)*x^3-(5*n+4)*x^4-(n^2+n)*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[((2 - (2*n - 16)*x - (3*n^2 - 21*n + 6)*x^2 - (n^3 - 9*n^2 + 8)*x^3 + (n^3 - n)*x^4)*P[n - 1, x] + (2*x + (3*n - 4)*x^2 + (n^2 - n - 10)*x^3 - (5*n + 4)*x^4 - (n^2 + n)*x^5)*D[P[n - 1, x], x])/(2 + 2*n*x + (n^2 - 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] = ((2 - (2*n - 16)*x - (3*n^2 - 21*n + 6)*x^2 - (n^3 - 9*n^2 + 8)*x^3 + (n^3 - n)*x^4)*v[i - 1] + (2*x + (3*n - 4)*x^2 + (n^2 - n - 10)*x^3 - (5*n + 4)*x^4 - (n^2 + n)*x^5)*deriv(v[i - 1], x))/(2 + 2*n*x + (n^2 - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.