Real-rooted atlas

A136011 → (θ+1)²

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2145
311213
4128938
516054115
6112422516366
712528101601223
8150827091040254283
9110208694560037515690
1012044272252721635003660022
11140928397012432026250756239389
1218188256509545680173775957649993778
13116380778734233200010631259525613724286868
14132764235462597840166168250821772226386419184130
Technical details
Node id
n-0a8cd4225ff87cee · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-eb678dbb44a51177

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,    4;
  3: 1,   12;
  4: 1,   28,    9;
  5: 1,   60,   54;
  6: 1,  124,  225,   16;
  7: 1,  252,  810,  160;
  8: 1,  508, 2709, 1040,  25;
  9: 1, 1020, 8694, 5600, 375;
%F Formula
The row polynomials satisfy P(n,x) = 9*x^2*P(n-4,x) + 4*x^3*d/dx P(n-4,x) - 4*x*P(n-3,x) + 4*x^2*d/dx P(n-3,x) + 4*x^3*d^2/dx^2 P(n-3,x) + 4*x*P(n-2,x) - (2*x+4*x^2)*d/dx P(n-2,x) - x^2*d^2/dx^2 P(n-2,x) + P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 4*x, P(3,x) = 1 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[3, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 4, x] + 4*x^3*D[P[n - 4, x], x] - 4*x*P[n - 3, x] + 4*x^2*D[P[n - 3, x], x] + 4*x^3*D[P[n - 3, x], {x, 2}] + 4*x*P[n - 2, x] - (2*x + 4*x^2)*D[P[n - 2, x], x] - x^2*D[P[n - 2, x], {x, 2}] + P[n - 1, x] + 2*x*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] = 1; v[3] = 1 + 4*x; v[4] = 1 + 12*x; for(n = 4, N, my(i = n + 1); v[i] = 9*x^2*v[i - 4] + 4*x^3*deriv(v[i - 4], x) - 4*x*v[i - 3] + 4*x^2*deriv(v[i - 3], x) + 4*x^3*deriv(deriv(v[i - 3], x), x) + 4*x*v[i - 2] - (2*x + 4*x^2)*deriv(v[i - 2], x) - x^2*deriv(deriv(v[i - 2], x), x) + v[i - 1] + 2*x*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. [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.