Real-rooted atlas

A390433 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-144 - 32\,x + 9\,x^{2} + 168\,n + 24\,n x - 73\,n^{2} - 4\,n^{2} x + 14\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(112\,x + 41\,x^{2} + 4\,x^{3} - 112\,n x - 24\,n x^{2} + 37\,n^{2} x + 4\,n^{2} x^{2} - 4\,n^{3} x\right)\,P_{n-2}'(x) + \left(-40\,x^{2} - 15\,x^{3} + 30\,n x^{2} + 8\,n x^{3} - 5\,n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(8\,x^{3} + 4\,x^{4} - 2\,n x^{3}\right)\,P_{n-2}'''(x) + \left(20 + 4\,x - 12\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(-11\,x - 4\,x^{2} + 4\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2044
3016925
4014418016340
5023044824896258049
605760017884852480300036291964
7020736008893152367667232500079204914976393
8010160640057382732831362048038244800142632017836641028743228

All 15 rows

Technical details
Node id
n-d7d91555d97d97ad · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-f5c62d1dc369412f

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0,            4;
   3: 0,           16,             9;
   4: 0,          144,           180,            16;
   5: 0,         2304,          4824,           896,           25;
   6: 0,        57600,        178848,         52480,         3000,          36;
   7: 0,      2073600,       8893152,       3676672,       325000,        7920,         49;
   8: 0,    101606400,     573827328,     313620480,     38244800,     1426320,      17836,       64;
   9: 0,   6502809600,   46708627968,   32382185472,   5117652800,   260462592,    4955664,    35840,    81;
  10: 0, 526727577600, 4685494118400, 4001282002944, 787539168000, 51387598080, 1325828672, 14536704, 66096, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-14*n^3+73*n^2-168*n+144+(4*n^2-24*n+32)*x-9*x^2)*P(n-2,x) - ((4*n^3-37*n^2+112*n-112)*x-(4*n^2-24*n+41)*x^2-4*x^3)*d/dx P(n-2,x) - ((5*n^2-30*n+40)*x^2-(8*n-15)*x^3)*d^2/dx^2 P(n-2,x) - ((2*n-8)*x^3-4*x^4)*d^3/dx^3 P(n-2,x) + (2*n^2-12*n+20+4*x)*P(n-1,x) + ((4*n-11)*x-4*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 14*n^3 + 73*n^2 - 168*n + 144 + (4*n^2 - 24*n + 32)*x - 9*x^2)*P[n - 2, x] - ((4*n^3 - 37*n^2 + 112*n - 112)*x - (4*n^2 - 24*n + 41)*x^2 - 4*x^3)*D[P[n - 2, x], x] - ((5*n^2 - 30*n + 40)*x^2 - (8*n - 15)*x^3)*D[P[n - 2, x], {x, 2}] - ((2*n - 8)*x^3 - 4*x^4)*D[P[n - 2, x], {x, 3}] + (2*n^2 - 12*n + 20 + 4*x)*P[n - 1, x] + ((4*n - 11)*x - 4*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; for(n = 3, N, my(i = n); v[i] = -(n^4 - 14*n^3 + 73*n^2 - 168*n + 144 + (4*n^2 - 24*n + 32)*x - 9*x^2)*v[i - 2] - ((4*n^3 - 37*n^2 + 112*n - 112)*x - (4*n^2 - 24*n + 41)*x^2 - 4*x^3)*deriv(v[i - 2], x) - ((5*n^2 - 30*n + 40)*x^2 - (8*n - 15)*x^3)*deriv(deriv(v[i - 2], x), x) - ((2*n - 8)*x^3 - 4*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2*n^2 - 12*n + 20 + 4*x)*v[i - 1] + ((4*n - 11)*x - 4*x^2)*deriv(v[i - 1], x) + 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, v[i])))};
T(10)

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.