Real-rooted atlas

A166962 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5\,x^{2} + 12\,n x^{2} - 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2} + 3\,x^{3} - 12\,n x^{2} + 9\,n x^{3}\right)\,P_{n-2}'(x) + \left(-3\,x^{3} - 9\,n x^{3}\right)\,P_{n-2}''(x) + \left(1 - 4\,x + 6\,n x\right)\,P_{n-1}(x) + \left(7\,x - 3\,x^{2}\right)\,P_{n-1}'(x) + \left(3\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
2124328
312062074418
4116686510126459448
511337016173011132066456293072
6110699236570456609200140597532472711811692
7185597479365753327756780174003375149275021524678597061860

All 15 rows

Technical details
Node id
n-8c87e3909ebe3753 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-49673c01b4703ab3

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,        2;
  2: 1,       24,           3;
  3: 1,      206,         207,            4;
  4: 1,     1668,        6510,         1264,             5;
  5: 1,    13370,      161730,       111320,          6645,            6;
  6: 1,   106992,     3657045,      6609200,       1405975,        32472,           7;
  7: 1,   855974,    79365753,    327756780,     174003375,     14927502,      152467,          8;
  8: 1,  6847836,  1691076072,  14803407264,   16636267050,   3521080692,   142186044,     698400,       9;
  9: 1, 54782738, 35738576100, 634976884384, 1377425503530, 597549512412, 59990018592, 1260406752, 3144825, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-12*n-5)*x^2*P(n-2,x) - ((12*n+4)*x^2-(9*n+3)*x^3)*d/dx P(n-2,x) - (9*n+3)*x^3*d^2/dx^2 P(n-2,x) + (1+(6*n-4)*x)*P(n-1,x) + (7*x-3*x^2)*d/dx P(n-1,x) + 3*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 12*n - 5)*x^2*P[n - 2, x] - ((12*n + 4)*x^2 - (9*n + 3)*x^3)*D[P[n - 2, x], x] - (9*n + 3)*x^3*D[P[n - 2, x], {x, 2}] + (1 + (6*n - 4)*x)*P[n - 1, x] + (7*x - 3*x^2)*D[P[n - 1, x], x] + 3*x^2*D[P[n - 1, x], {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; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 12*n - 5)*x^2*v[i - 2] - ((12*n + 4)*x^2 - (9*n + 3)*x^3)*deriv(v[i - 2], x) - (9*n + 3)*x^3*deriv(deriv(v[i - 2], x), x) + (1 + (6*n - 4)*x)*v[i - 1] + (7*x - 3*x^2)*deriv(v[i - 1], x) + 3*x^2*deriv(deriv(v[i - 1], x), 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.