Real-rooted atlas

A156289 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(39\,x^{2}\right)\,P_{n-2}(x) + \left(13\,x^{2} + 56\,x^{3}\right)\,P_{n-2}'(x) + \left(25\,x^{3} + 12\,x^{4}\right)\,P_{n-2}''(x) + \left(6\,x^{4}\right)\,P_{n-2}'''(x) + \left(1 + 12\,x\right)\,P_{n-1}(x) + \left(3\,x - 4\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211213
3160135196
41252189016803823
51102019845504002362594891
614092190080105336012993753742202921128
711638017567551921920047297250340540206621615108965221
816553215995070329369040145270125019751331609270261001297296004830019753

All 15 rows

Technical details
Node id
n-90a8e43a61417d9f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-dc5794018258dfe5

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: 1,      12;
   3: 1,      60,        135;
   4: 1,     252,       1890,        1680;
   5: 1,    1020,      19845,       50400,         23625;
   6: 1,    4092,     190080,     1053360,       1299375,        374220;
   7: 1,   16380,    1756755,    19219200,      47297250,      34054020,       6621615;
   8: 1,   65532,   15995070,   329369040,    1452701250,    1975133160,     927026100,     129729600;
   9: 1,  262140,  144692865,  5468954400,   40949283375,   94115581560,   80373162870,   26464838400,   2791213425;
  10: 1, 1048572, 1305184860, 89303892720, 1100639255625, 4036797584820, 5603607909900, 3268407542400, 795495826125, 65472907500;
%F Formula
The row polynomials satisfy P(n,x) = 39*x^2*P(n-2,x) + (13*x^2+56*x^3)*d/dx P(n-2,x) + (25*x^3+12*x^4)*d^2/dx^2 P(n-2,x) + 6*x^4*d^3/dx^3 P(n-2,x) + (1+12*x)*P(n-1,x) + (3*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) = 1 + 12*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[39*x^2*P[n - 2, x] + (13*x^2 + 56*x^3)*D[P[n - 2, x], x] + (25*x^3 + 12*x^4)*D[P[n - 2, x], {x, 2}] + 6*x^4*D[P[n - 2, x], {x, 3}] + (1 + 12*x)*P[n - 1, x] + (3*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] = 1 + 12*x; for(n = 3, N, my(i = n); v[i] = 39*x^2*v[i - 2] + (13*x^2 + 56*x^3)*deriv(v[i - 2], x) + (25*x^3 + 12*x^4)*deriv(deriv(v[i - 2], x), x) + 6*x^4*deriv(deriv(deriv(v[i - 2], x), x), x) + (1 + 12*x)*v[i - 1] + (3*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.