Real-rooted atlas

A143494 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 10\,x + 5\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(2\,x + 3\,x^{2} + x^{3}\right)\,P_{n-3}'(x) + \left(-11 - 11\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(-3\,x - 2\,x^{2}\right)\,P_{n-2}'(x) + \left(6 + 3\,x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)
\(P_{3}(x) = 14 + 16\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2426
31416333
44692394181
51464603337651020
64542132235590013065977
7139494361497385402035204736589
842464053289019712042492040743018234304

All 15 rows

Technical details
Node id
n-8719fd1736db79b1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-aa827404435f1c6b

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,      2;
   3:    14,     16,       3;
   4:    46,     92,      39,       4;
   5:   146,    460,     333,      76,       5;
   6:   454,   2132,    2355,     900,     130,      6;
   7:  1394,   9436,   14973,    8540,    2035,    204,      7;
   8:  4246,  40532,   89019,   71204,   24920,   4074,    301,     8;
   9: 12866, 170620,  505893,  545916,  263445,  62496,   7462,   424,   9;
  10: 38854, 708212, 2785395, 3950020, 2526510, 816102, 140070, 12768, 576, 10;
%F Formula
The row polynomials satisfy P(n,x) = (6+10*x+5*x^2+x^3)*P(n-3,x) + (2*x+3*x^2+x^3)*d/dx P(n-3,x) - (11+11*x+3*x^2)*P(n-2,x) - (3*x+2*x^2)*d/dx P(n-2,x) + (6+3*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 4 + 2*x, P(3,x) = 14 + 16*x + 3*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[3, x_] := 14 + 16*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 10*x + 5*x^2 + x^3)*P[n - 3, x] + (2*x + 3*x^2 + x^3)*D[P[n - 3, x], x] - (11 + 11*x + 3*x^2)*P[n - 2, x] - (3*x + 2*x^2)*D[P[n - 2, x], x] + (6 + 3*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; v[3] = 14 + 16*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (6 + 10*x + 5*x^2 + x^3)*v[i - 3] + (2*x + 3*x^2 + x^3)*deriv(v[i - 3], x) - (11 + 11*x + 3*x^2)*v[i - 2] - (3*x + 2*x^2)*deriv(v[i - 2], x) + (6 + 3*x)*v[i - 1] + 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, 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. [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
  4. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.