Real-rooted atlas

A199881 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9\,x^{2}\right)\,P_{n-4}(x) + \left(4\,x^{3}\right)\,P_{n-4}'(x) + \left(18\,x^{2}\right)\,P_{n-3}(x) + \left(8\,x^{3}\right)\,P_{n-3}'(x) + \left(4\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(4\,x\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 4\,x\)
\(P_{1}(x) = 4\,x + 9\,x^{2}\)
\(P_{2}(x) = 4\,x + 18\,x^{2} + 16\,x^{3}\)
\(P_{3}(x) = 18\,x^{2} + 48\,x^{3} + 25\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0145
104913
204181638
30018482591
40096410036209
50004817518049452
60001617539629464945
70000100504784448811917
8000025396122514086481003802
9000001801225262423499001217399
100000036784320051033700121014414177
110000002942624737192005566158416926808
12000000491408737115400156098064202819650125
130000000448510318200297662520011323254822592813
14000000064234915400406565400039039154843150256170398
Technical details
Node id
n-d5ece7e55178ede2 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c5d4b77a0d1a232e

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1, 4;
  1: 0, 4,  9;
  2: 0, 4, 18, 16;
  3: 0, 0, 18, 48,  25;
  4: 0, 0,  9, 64, 100,  36;
  5: 0, 0,  0, 48, 175, 180,   49;
  6: 0, 0,  0, 16, 175, 396,  294,   64;
  7: 0, 0,  0,  0, 100, 504,  784,  448,   81;
  8: 0, 0,  0,  0,  25, 396, 1225, 1408,  648, 100;
  9: 0, 0,  0,  0,   0, 180, 1225, 2624, 2349, 900, 121;
%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) + 18*x^2*P(n-3,x) + 8*x^3*d/dx P(n-3,x) + (4*x+9*x^2)*P(n-2,x) - (4*x^2-4*x^3)*d/dx P(n-2,x) + 4*x*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1 + 4*x, P(1,x) = 4*x + 9*x^2, P(2,x) = 4*x + 18*x^2 + 16*x^3, P(3,x) = 18*x^2 + 48*x^3 + 25*x^4.
%t Mathematica
P[0, x_] := 1 + 4*x;
P[1, x_] := 4*x + 9*x^2;
P[2, x_] := 4*x + 18*x^2 + 16*x^3;
P[3, x_] := 18*x^2 + 48*x^3 + 25*x^4;
P[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 4, x] + 4*x^3*D[P[n - 4, x], x] + 18*x^2*P[n - 3, x] + 8*x^3*D[P[n - 3, x], x] + (4*x + 9*x^2)*P[n - 2, x] - (4*x^2 - 4*x^3)*D[P[n - 2, x], x] + 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 + 4*x; v[2] = 4*x + 9*x^2; v[3] = 4*x + 18*x^2 + 16*x^3; v[4] = 18*x^2 + 48*x^3 + 25*x^4; for(n = 4, N, my(i = n + 1); v[i] = 9*x^2*v[i - 4] + 4*x^3*deriv(v[i - 4], x) + 18*x^2*v[i - 3] + 8*x^3*deriv(v[i - 3], x) + (4*x + 9*x^2)*v[i - 2] - (4*x^2 - 4*x^3)*deriv(v[i - 2], x) + 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. [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.