Real-rooted atlas

A063967 → (θ+1)²

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(25 - 85\,x + 34\,x^{2} + 26\,x^{3} + x^{4} - x^{5} + 75\,n x - 110\,n x^{2} - 18\,n x^{3} + 14\,n x^{4} + n x^{5} + 60\,n^{2} x^{2} - 25\,n^{2} x^{3} - 13\,n^{2} x^{4} + n^{2} x^{5} + 15\,n^{3} x^{3} - n^{3} x^{5} + n^{4} x^{4}\right)\,P_{n}(x) = \left(25 + 15\,x - 26\,x^{2} - 18\,x^{3} + x^{4} + 3\,x^{5} + 75\,n x + 85\,n x^{2} - 13\,n x^{3} - 31\,n x^{4} - 8\,n x^{5} + 60\,n^{2} x^{2} + 80\,n^{2} x^{3} + 13\,n^{2} x^{4} - 9\,n^{2} x^{5} - 2\,n^{2} x^{6} + 15\,n^{3} x^{3} + 19\,n^{3} x^{4} + 3\,n^{3} x^{5} - n^{3} x^{6} + n^{4} x^{4} + n^{4} x^{5}\right)\,P_{n-2}(x) + \left(25 + 15\,x - 26\,x^{2} - 18\,x^{3} + x^{4} + 3\,x^{5} + 75\,n x + 50\,n x^{2} - 68\,n x^{3} - 48\,n x^{4} - 5\,n x^{5} + 60\,n^{2} x^{2} + 65\,n^{2} x^{3} - 6\,n^{2} x^{4} - 12\,n^{2} x^{5} - n^{2} x^{6} + 15\,n^{3} x^{3} + 17\,n^{3} x^{4} + n^{3} x^{5} - n^{3} x^{6} + n^{4} x^{4} + n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2145
3212923
4328451692
556014411225346
68120387464225361240
71323293615201150396494296
82143621154336447524126376414496

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-8d4eedf9d66a931a · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-bd81fa9c8482d6ca

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  1,    4;
   3:  2,   12,    9;
   4:  3,   28,   45,    16;
   5:  5,   60,  144,   112,    25;
   6:  8,  120,  387,   464,   225,    36;
   7: 13,  232,  936,  1520,  1150,   396,    49;
   8: 21,  436, 2115,  4336,  4475,  2412,   637,   64;
   9: 34,  804, 4554, 11280, 14775, 10908,  4508,  960,   81;
  10: 55, 1460, 9459, 27472, 43650, 41040, 23275, 7744, 1377, 100;
%F Formula
The row polynomials satisfy (25+(75*n-85)*x+(60*n^2-110*n+34)*x^2+(15*n^3-25*n^2-18*n+26)*x^3+(n^4-13*n^2+14*n+1)*x^4-(n^3-n^2-n+1)*x^5)*P(n,x) = (25+(75*n+15)*x+(60*n^2+85*n-26)*x^2+(15*n^3+80*n^2-13*n-18)*x^3+(n^4+19*n^3+13*n^2-31*n+1)*x^4+(n^4+3*n^3-9*n^2-8*n+3)*x^5-(n^3+2*n^2)*x^6)*P(n-2,x) + (25+(75*n+15)*x+(60*n^2+50*n-26)*x^2+(15*n^3+65*n^2-68*n-18)*x^3+(n^4+17*n^3-6*n^2-48*n+1)*x^4+(n^4+n^3-12*n^2-5*n+3)*x^5-(n^3+n^2)*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((25 + (75*n + 15)*x + (60*n^2 + 85*n - 26)*x^2 + (15*n^3 + 80*n^2 - 13*n - 18)*x^3 + (n^4 + 19*n^3 + 13*n^2 - 31*n + 1)*x^4 + (n^4 + 3*n^3 - 9*n^2 - 8*n + 3)*x^5 - (n^3 + 2*n^2)*x^6)*P[n - 2, x] + (25 + (75*n + 15)*x + (60*n^2 + 50*n - 26)*x^2 + (15*n^3 + 65*n^2 - 68*n - 18)*x^3 + (n^4 + 17*n^3 - 6*n^2 - 48*n + 1)*x^4 + (n^4 + n^3 - 12*n^2 - 5*n + 3)*x^5 - (n^3 + n^2)*x^6)*P[n - 1, x])/(25 + (75*n - 85)*x + (60*n^2 - 110*n + 34)*x^2 + (15*n^3 - 25*n^2 - 18*n + 26)*x^3 + (n^4 - 13*n^2 + 14*n + 1)*x^4 - (n^3 - n^2 - n + 1)*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = ((25 + (75*n + 15)*x + (60*n^2 + 85*n - 26)*x^2 + (15*n^3 + 80*n^2 - 13*n - 18)*x^3 + (n^4 + 19*n^3 + 13*n^2 - 31*n + 1)*x^4 + (n^4 + 3*n^3 - 9*n^2 - 8*n + 3)*x^5 - (n^3 + 2*n^2)*x^6)*v[i - 2] + (25 + (75*n + 15)*x + (60*n^2 + 50*n - 26)*x^2 + (15*n^3 + 65*n^2 - 68*n - 18)*x^3 + (n^4 + 17*n^3 - 6*n^2 - 48*n + 1)*x^4 + (n^4 + n^3 - 12*n^2 - 5*n + 3)*x^5 - (n^3 + n^2)*x^6)*v[i - 1])/(25 + (75*n - 85)*x + (60*n^2 - 110*n + 34)*x^2 + (15*n^3 - 25*n^2 - 18*n + 26)*x^3 + (n^4 - 13*n^2 + 14*n + 1)*x^4 - (n^3 - n^2 - n + 1)*x^5)); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-aa7e9b3181553556
  • Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-aa7e9b3181553556
  • Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-aa7e9b3181553556

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.