Real-rooted atlas

A228955 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 8\,x + 4\,x^{2} - 42\,n x + 12\,n x^{2} - 22\,n^{2} x + 13\,n^{2} x^{2} + 6\,n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n}(x) = \left(36\,n + 136\,n x - 104\,n x^{2} + 36\,n x^{3} - 42\,n^{2} x - 130\,n^{2} x^{2} + 60\,n^{2} x^{3} - 22\,n^{3} x - 25\,n^{3} x^{2} + 37\,n^{3} x^{3} + 6\,n^{4} x^{2} + 10\,n^{4} x^{3} + n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(72\,x - 216\,x^{2} + 216\,x^{3} - 72\,x^{4} - 172\,n x^{2} + 292\,n x^{3} - 120\,n x^{4} - 44\,n^{2} x^{2} + 118\,n^{2} x^{3} - 74\,n^{2} x^{4} + 20\,n^{3} x^{3} - 20\,n^{3} x^{4} + 2\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4Pn(1)
011
1145
222426
3614454204
42496010802064
5120720016200192025440
67206048022680080640368640
75040564480317520022579201260006128640

All 15 rows

Technical details
Node id
n-6ec34513293cdda0 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1f0181c04d99414a

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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;
  1:      1,        4;
  2:      2,       24;
  3:      6,      144,        54;
  4:     24,      960,      1080;
  5:    120,     7200,     16200,       1920;
  6:    720,    60480,    226800,      80640;
  7:   5040,   564480,   3175200,    2257920,    126000;
  8:  40320,  5806080,  45722880,   54190080,   9072000;
  9: 362880, 65318400, 685843200, 1219276800, 408240000, 13063680;
%F Formula
The row polynomials satisfy (36-(22*n^2+42*n+8)*x+(n^4+6*n^3+13*n^2+12*n+4)*x^2)*P(n,x) = (36*n-(22*n^3+42*n^2-136*n)*x+(n^5+6*n^4-25*n^3-130*n^2-104*n)*x^2+(n^5+10*n^4+37*n^3+60*n^2+36*n)*x^3)*P(n-1,x) + (72*x-(44*n^2+172*n+216)*x^2+(2*n^4+20*n^3+118*n^2+292*n+216)*x^3-(2*n^4+20*n^3+74*n^2+120*n+72)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((36*n - (22*n^3 + 42*n^2 - 136*n)*x + (n^5 + 6*n^4 - 25*n^3 - 130*n^2 - 104*n)*x^2 + (n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n)*x^3)*P[n - 1, x] + (72*x - (44*n^2 + 172*n + 216)*x^2 + (2*n^4 + 20*n^3 + 118*n^2 + 292*n + 216)*x^3 - (2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x^4)*D[P[n - 1, x], x])/(36 - (22*n^2 + 42*n + 8)*x + (n^4 + 6*n^3 + 13*n^2 + 12*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((36*n - (22*n^3 + 42*n^2 - 136*n)*x + (n^5 + 6*n^4 - 25*n^3 - 130*n^2 - 104*n)*x^2 + (n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n)*x^3)*v[i - 1] + (72*x - (44*n^2 + 172*n + 216)*x^2 + (2*n^4 + 20*n^3 + 118*n^2 + 292*n + 216)*x^3 - (2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x^4)*deriv(v[i - 1], x))/(36 - (22*n^2 + 42*n + 8)*x + (n^4 + 6*n^3 + 13*n^2 + 12*n + 4)*x^2)); 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

Same fingerprint ?

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.