Real-rooted atlas

A007318 → (1/2+(1/2)n)θ+1+n → θ+1

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(-1 - 4\,x - 6\,x^{2} - 4\,x^{3} - x^{4}\right)\,P_{n-4}(x) + \left(4 + 12\,x + 12\,x^{2} + 4\,x^{3}\right)\,P_{n-3}(x) + \left(-6 - 12\,x - 6\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 6\,x\)
\(P_{2}(x) = 3 + 18\,x + 18\,x^{2}\)
\(P_{3}(x) = 4 + 36\,x + 72\,x^{2} + 40\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1268
23181839
34367240152
456018020075520
56903606004501261632
67126630140015758821964816
781681008280042003528156828813568

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-f040903628c87c74 · revision 1 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5dbc84ad685f9f2d

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 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:  2,   6;
  2:  3,  18,   18;
  3:  4,  36,   72,   40;
  4:  5,  60,  180,  200,    75;
  5:  6,  90,  360,  600,   450,   126;
  6:  7, 126,  630, 1400,  1575,   882,   196;
  7:  8, 168, 1008, 2800,  4200,  3528,  1568,   288;
  8:  9, 216, 1512, 5040,  9450, 10584,  7056,  2592,  405;
  9: 10, 270, 2160, 8400, 18900, 26460, 23520, 12960, 4050, 550;
%F Formula
The row polynomials satisfy P(n,x) = -(1+4*x+6*x^2+4*x^3+x^4)*P(n-4,x) + (4+12*x+12*x^2+4*x^3)*P(n-3,x) - (6+12*x+6*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 6*x, P(2,x) = 3 + 18*x + 18*x^2, P(3,x) = 4 + 36*x + 72*x^2 + 40*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[2, x_] := 3 + 18*x + 18*x^2;
P[3, x_] := 4 + 36*x + 72*x^2 + 40*x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*P[n - 4, x] + (4 + 12*x + 12*x^2 + 4*x^3)*P[n - 3, x] - (6 + 12*x + 6*x^2)*P[n - 2, x] + (4 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 6*x; v[3] = 3 + 18*x + 18*x^2; v[4] = 4 + 36*x + 72*x^2 + 40*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*v[i - 4] + (4 + 12*x + 12*x^2 + 4*x^3)*v[i - 3] - (6 + 12*x + 6*x^2)*v[i - 2] + (4 + 4*x)*v[i - 1]); 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-cebd9f84185370dc
  • 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-cebd9f84185370dc
  • 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-cebd9f84185370dc

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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  5. [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
  6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.