Real-rooted atlas

A088996 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n - 12\,n x + 9\,n x^{2} + 12\,n^{2} + 20\,n^{2} x - 18\,n^{2} x^{2} - 4\,n^{3} - 8\,n^{3} x + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(12\,n x^{2} + 4\,n x^{3} - 20\,n^{2} x^{2} - 8\,n^{2} x^{3} + 8\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(1 - 6\,n - 4\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
23201841
31513622596472
41051192279922726006968
5945128283860145328231504320125172
610395163788597789910352679600247536352802644740
7135135242030410350918192737281879177510068336281710832256064179864

All 15 rows

Technical details
Node id
n-622d91c75ec0122a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-751cc9a161a29d72

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,         4;
  2:        3,        20,         18;
  3:       15,       136,        225,          96;
  4:      105,      1192,       2799,        2272,         600;
  5:      945,     12828,      38601,       45328,       23150,        4320;
  6:    10395,    163788,     597789,      910352,      679600,      247536,      35280;
  7:   135135,   2420304,   10350918,    19273728,    18791775,    10068336,    2817108,     322560;
  8:  2027025,  40628880,  198829242,   436318976,   522798225,   367506288,  151889612,   34274304,   3265920;
  9: 34459425, 763663860, 4202831934, 10598690464, 15023305425, 13023071892, 7084075432, 2368141056, 445926384, 36288000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(8*n^3-20*n^2+12*n)*x-(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + ((8*n^3-20*n^2+12*n)*x^2+(4*n^3-8*n^2+4*n)*x^3)*d/dx P(n-2,x) + (4*n^2-6*n+1+(4*n^2-4*n)*x)*P(n-1,x) - (4*n^2-4*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (8*n^3 - 20*n^2 + 12*n)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + ((8*n^3 - 20*n^2 + 12*n)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 6*n + 1 + (4*n^2 - 4*n)*x)*P[n - 1, x] - (4*n^2 - 4*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];
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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (8*n^3 - 20*n^2 + 12*n)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + ((8*n^3 - 20*n^2 + 12*n)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 6*n + 1 + (4*n^2 - 4*n)*x)*v[i - 1] - (4*n^2 - 4*n)*x^2*deriv(v[i - 1], x))/(n - 1)); 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.