Real-rooted atlas

A188440 → (θ+1)²

nonnegative coefficients

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

Recurrence
\(\left(4 + 7\,x + x^{2} - 6\,x^{3} + 10\,n x + 6\,n x^{2} - 3\,n x^{3} + 14\,n^{2} x^{2} + 11\,n^{2} x^{3} + n^{2} x^{4} + 6\,n^{3} x^{3} + 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(4 - 25\,x - 55\,x^{2} - 58\,x^{3} - 20\,x^{4} - 8\,x^{5} + 42\,n x - 18\,n x^{2} - 69\,n x^{3} - 62\,n x^{4} - 16\,n x^{5} + 94\,n^{2} x^{2} + 63\,n^{2} x^{3} + 9\,n^{2} x^{4} - 2\,n^{2} x^{5} + 72\,n^{3} x^{3} + 56\,n^{3} x^{4} + 14\,n^{3} x^{5} + 21\,n^{4} x^{4} + 10\,n^{4} x^{5} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-8\,x - 36\,x^{2} - 68\,x^{3} - 76\,x^{4} - 48\,x^{5} - 16\,x^{6} - 34\,n x^{2} - 138\,n x^{3} - 206\,n x^{4} - 156\,n x^{5} - 48\,n x^{6} - 40\,n^{2} x^{3} - 138\,n^{2} x^{4} - 142\,n^{2} x^{5} - 52\,n^{2} x^{6} - 16\,n^{3} x^{4} - 44\,n^{3} x^{5} - 24\,n^{3} x^{6} - 2\,n^{4} x^{5} - 4\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2189
3189
41163653
51163653
6124108128261
7124108128261
81322165124001161
91322165124001161
101403601280200011524833
111403601280200011524833
12148540256060006912313619197
13148540256060006912313619197
141567564480140002419221952819273629

Computed in your browser ?

Technical details
Node id
n-72ed2075ed1a9d58 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-5367d60e6d4311ec

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1,  8;
  3: 1,  8;
  4: 1, 16,  36;
  5: 1, 16,  36;
  6: 1, 24, 108, 128;
  7: 1, 24, 108, 128;
  8: 1, 32, 216, 512, 400;
  9: 1, 32, 216, 512, 400;
%F Formula
The row polynomials satisfy (4+(10*n+7)*x+(14*n^2+6*n+1)*x^2+(6*n^3+11*n^2-3*n-6)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (4+(42*n-25)*x+(94*n^2-18*n-55)*x^2+(72*n^3+63*n^2-69*n-58)*x^3+(21*n^4+56*n^3+9*n^2-62*n-20)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) - (8*x+(34*n+36)*x^2+(40*n^2+138*n+68)*x^3+(16*n^3+138*n^2+206*n+76)*x^4+(2*n^4+44*n^3+142*n^2+156*n+48)*x^5+(4*n^4+24*n^3+52*n^2+48*n+16)*x^6)*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[((4 + (42*n - 25)*x + (94*n^2 - 18*n - 55)*x^2 + (72*n^3 + 63*n^2 - 69*n - 58)*x^3 + (21*n^4 + 56*n^3 + 9*n^2 - 62*n - 20)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] - (8*x + (34*n + 36)*x^2 + (40*n^2 + 138*n + 68)*x^3 + (16*n^3 + 138*n^2 + 206*n + 76)*x^4 + (2*n^4 + 44*n^3 + 142*n^2 + 156*n + 48)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(4 + (10*n + 7)*x + (14*n^2 + 6*n + 1)*x^2 + (6*n^3 + 11*n^2 - 3*n - 6)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)]];
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] = ((4 + (42*n - 25)*x + (94*n^2 - 18*n - 55)*x^2 + (72*n^3 + 63*n^2 - 69*n - 58)*x^3 + (21*n^4 + 56*n^3 + 9*n^2 - 62*n - 20)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] - (8*x + (34*n + 36)*x^2 + (40*n^2 + 138*n + 68)*x^3 + (16*n^3 + 138*n^2 + 206*n + 76)*x^4 + (2*n^4 + 44*n^3 + 142*n^2 + 156*n + 48)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(4 + (10*n + 7)*x + (14*n^2 + 6*n + 1)*x^2 + (6*n^3 + 11*n^2 - 3*n - 6)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, 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-0d38730cf56d64c0
  • Coefficients of one sign. Every nonzero row, n >= 0: 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-0d38730cf56d64c0
  • Hadamard kernel (Malo). Rows P_n, n >= 0, 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-0d38730cf56d64c0

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

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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