Real-rooted atlas

A269947 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-64 + 32\,x + 9\,x^{2} + 192\,n - 48\,n x - 240\,n^{2} + 24\,n^{2} x + 160\,n^{3} - 4\,n^{3} x - 60\,n^{4} + 12\,n^{5} - n^{6}\right)\,P_{n-2}(x) + \left(-32\,x^{2} + 4\,x^{3} + 48\,n x^{2} - 24\,n^{2} x^{2} + 4\,n^{3} x^{2}\right)\,P_{n-2}'(x) + \left(-9 + 4\,x + 15\,n - 9\,n^{2} + 2\,n^{3}\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1044
204913
30328116129
408642259576253724
505529614652040880250036245232
60691200018439416537048037637581004931106420
7014929920003998465856119280486489688375229158021609646776264348

All 15 rows

Technical details
Node id
n-78b283ef74bca0e9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-77a03caf486e7d7a

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: 0,               4;
  2: 0,               4,               9;
  3: 0,              32,              81,              16;
  4: 0,             864,            2259,             576,             25;
  5: 0,           55296,          146520,           40880,           2500,           36;
  6: 0,         6912000,        18439416,         5370480,         376375,         8100,         49;
  7: 0,      1492992000,      3998465856,      1192804864,       89688375,      2291580,      21609,       64;
  8: 0,    512096256000,   1374833020608,    416240452096,    32626870225,    915163200,   10530982,    50176,     81;
  9: 0, 262193283072000, 705066723127296, 215559259065344, 17355333261600, 515546251524, 6637501584, 39444864, 104976, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(n^6-12*n^5+60*n^4-160*n^3+240*n^2-192*n+64+(4*n^3-24*n^2+48*n-32)*x-9*x^2)*P(n-2,x) + ((4*n^3-24*n^2+48*n-32)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n^3-9*n^2+15*n-9+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (4*n^3 - 24*n^2 + 48*n - 32)*x - 9*x^2)*P[n - 2, x] + ((4*n^3 - 24*n^2 + 48*n - 32)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n^3 - 9*n^2 + 15*n - 9 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];
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] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (4*n^3 - 24*n^2 + 48*n - 32)*x - 9*x^2)*v[i - 2] + ((4*n^3 - 24*n^2 + 48*n - 32)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n^3 - 9*n^2 + 15*n - 9 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], 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.