Real-rooted atlas

A008299 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-9\,n x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-4}(x) + \left(-4\,n x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-4}'(x) + \left(-4\,n x - 4\,n^{2} x\right)\,P_{n-3}(x) + \left(4\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n-3}'(x) + \left(4\,n x^{3} + 4\,n^{2} x^{3}\right)\,P_{n-3}''(x) + \left(4\,n x + 4\,n^{2} x\right)\,P_{n-2}(x) + \left(-2\,x - 2\,n x - 4\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(-x^{2} - n x^{2}\right)\,P_{n-2}''(x) + \left(n\right)\,P_{n-1}(x) + \left(x + 2\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 12\,x\)
\(P_{3}(x) = 1 + 40\,x\)

Rows ?

nx0x1x2x3Pn(1)
011
111
211213
314041
41100135236
512249451170
61476441016806567
71984172622016038407

All 15 rows

Technical details
Node id
n-e570d9ba6500c5b4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7c3deb0bac03e7c1

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: 1,   12;
  3: 1,   40;
  4: 1,  100,    135;
  5: 1,  224,    945;
  6: 1,  476,   4410,   1680;
  7: 1,  984,  17262,  20160;
  8: 1, 2004,  61425, 151200,  23625;
  9: 1, 4048, 206415, 911680, 433125;
%F Formula
The row polynomials satisfy n*P(n,x) = (9*n^3-9*n)*x^2*P(n-4,x) + (4*n^3-4*n)*x^3*d/dx P(n-4,x) - (4*n^2+4*n)*x*P(n-3,x) + (4*n^2+4*n)*x^2*d/dx P(n-3,x) + (4*n^2+4*n)*x^3*d^2/dx^2 P(n-3,x) + (4*n^2+4*n)*x*P(n-2,x) - ((2*n+2)*x+(4*n^2+4*n)*x^2)*d/dx P(n-2,x) - (n+1)*x^2*d^2/dx^2 P(n-2,x) + n*P(n-1,x) + (2*n+1)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 12*x, P(3,x) = 1 + 40*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1 + 12*x;
P[3, x_] := 1 + 40*x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 9*n)*x^2*P[n - 4, x] + (4*n^3 - 4*n)*x^3*D[P[n - 4, x], x] - (4*n^2 + 4*n)*x*P[n - 3, x] + (4*n^2 + 4*n)*x^2*D[P[n - 3, x], x] + (4*n^2 + 4*n)*x^3*D[P[n - 3, x], {x, 2}] + (4*n^2 + 4*n)*x*P[n - 2, x] - ((2*n + 2)*x + (4*n^2 + 4*n)*x^2)*D[P[n - 2, x], x] - (n + 1)*x^2*D[P[n - 2, x], {x, 2}] + n*P[n - 1, x] + (2*n + 1)*x*D[P[n - 1, x], x])/(n)]];
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; v[3] = 1 + 12*x; v[4] = 1 + 40*x; for(n = 4, N, my(i = n + 1); v[i] = ((9*n^3 - 9*n)*x^2*v[i - 4] + (4*n^3 - 4*n)*x^3*deriv(v[i - 4], x) - (4*n^2 + 4*n)*x*v[i - 3] + (4*n^2 + 4*n)*x^2*deriv(v[i - 3], x) + (4*n^2 + 4*n)*x^3*deriv(deriv(v[i - 3], x), x) + (4*n^2 + 4*n)*x*v[i - 2] - ((2*n + 2)*x + (4*n^2 + 4*n)*x^2)*deriv(v[i - 2], x) - (n + 1)*x^2*deriv(deriv(v[i - 2], x), x) + n*v[i - 1] + (2*n + 1)*x*deriv(v[i - 1], x))/(n)); 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.