Real-rooted atlas

A281000 → D+1

nonnegative coefficients

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

Recurrence

\(\left(8\,n - 8\,n x + n x^{2} + 13\,n^{2} - 14\,n^{2} x + 2\,n^{2} x^{2} - 5\,n^{3} + 4\,n^{3} x + 2\,n^{4}\right)\,P_{n}(x) = \left(-12 + 20\,x - 16\,x^{2} + 2\,x^{3} + 2\,n - 28\,n x + 60\,n x^{2} - 8\,n x^{3} + 46\,n^{2} - 28\,n^{2} x - 50\,n^{2} x^{2} + 6\,n^{2} x^{3} - 8\,n^{3} - 6\,n^{3} x + 12\,n^{3} x^{2} + 8\,n^{4} + 6\,n^{4} x\right)\,P_{n-1}(x) + \left(-8\,x + 26\,x^{2} - 10\,x^{3} + x^{4} + 44\,n x - 115\,n x^{2} + 42\,n x^{3} - 4\,n x^{4} - 28\,n^{2} x + 39\,n^{2} x^{2} - 8\,n^{2} x^{3} + 16\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2415
32012133
4105112241242
55469243604011871
627726930429088060114933
7137284804843758143001820841121739
86649531231239639619019038675336011211007541

All 15 rows

Technical details
Node id
n-b44d3004ad6d24c5 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9eda192efee06871

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:       4,        1;
   3:      20,       12,        1;
   4:     105,      112,       24,        1;
   5:     546,      924,      360,       40,       1;
   6:    2772,     6930,     4290,      880,      60,       1;
   7:   13728,    48048,    43758,    14300,    1820,      84,      1;
   8:   66495,   312312,   396396,   190190,   38675,    3360,    112,    1;
   9:  316030,  1925352,  3267264,  2178176,  649740,   90440,   5712,  144,   1;
  10: 1478048, 11362494, 24942060, 22170720, 9170616, 1872108, 189924, 9120, 180, 1;
%F Formula
The row polynomials satisfy (2*n^4-5*n^3+13*n^2+8*n+(4*n^3-14*n^2-8*n)*x+(2*n^2+n)*x^2)*P(n,x) = (8*n^4-8*n^3+46*n^2+2*n-12+(6*n^4-6*n^3-28*n^2-28*n+20)*x+(12*n^3-50*n^2+60*n-16)*x^2+(6*n^2-8*n+2)*x^3)*P(n-1,x) + ((16*n^3-28*n^2+44*n-8)*x-(4*n^3-39*n^2+115*n-26)*x^2-(8*n^2-42*n+10)*x^3-(4*n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^4 - 8*n^3 + 46*n^2 + 2*n - 12 + (6*n^4 - 6*n^3 - 28*n^2 - 28*n + 20)*x + (12*n^3 - 50*n^2 + 60*n - 16)*x^2 + (6*n^2 - 8*n + 2)*x^3)*P[n - 1, x] + ((16*n^3 - 28*n^2 + 44*n - 8)*x - (4*n^3 - 39*n^2 + 115*n - 26)*x^2 - (8*n^2 - 42*n + 10)*x^3 - (4*n - 1)*x^4)*D[P[n - 1, x], x])/(2*n^4 - 5*n^3 + 13*n^2 + 8*n + (4*n^3 - 14*n^2 - 8*n)*x + (2*n^2 + n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^4 - 8*n^3 + 46*n^2 + 2*n - 12 + (6*n^4 - 6*n^3 - 28*n^2 - 28*n + 20)*x + (12*n^3 - 50*n^2 + 60*n - 16)*x^2 + (6*n^2 - 8*n + 2)*x^3)*v[i - 1] + ((16*n^3 - 28*n^2 + 44*n - 8)*x - (4*n^3 - 39*n^2 + 115*n - 26)*x^2 - (8*n^2 - 42*n + 10)*x^3 - (4*n - 1)*x^4)*deriv(v[i - 1], x))/(2*n^4 - 5*n^3 + 13*n^2 + 8*n + (4*n^3 - 14*n^2 - 8*n)*x + (2*n^2 + n)*x^2)); 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. [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.