Real-rooted atlas

A038220 → (θ+1)²

nonnegative coefficients

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

Recurrence
\(\left(9 - 6\,x + 18\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(27 + 54\,x + 36\,x^{2} + 8\,x^{3} + 54\,n x + 60\,n x^{2} + 16\,n x^{3} + 12\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13811
29483693
327216324128695
481864194415364004825
524332409720115206000115231875
672911664437406912054000207363136203125
72187408241837083628803780002177286585681921259375

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-0ee65094b134f8b2 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e543af4f498eab8b

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:     3,      8;
  2:     9,     48,      36;
  3:    27,    216,     324,     128;
  4:    81,    864,    1944,    1536,      400;
  5:   243,   3240,    9720,   11520,     6000,     1152;
  6:   729,  11664,   43740,   69120,    54000,    20736,    3136;
  7:  2187,  40824,  183708,  362880,   378000,   217728,   65856,    8192;
  8:  6561, 139968,  734832, 1741824,  2268000,  1741824,  790272,  196608,  20736;
  9: 19683, 472392, 2834352, 7838208, 12247200, 11757312, 7112448, 2654208, 559872, 51200;
%F Formula
The row polynomials satisfy (9+(18*n-6)*x+4*n^2*x^2)*P(n,x) = (27+(54*n+54)*x+(12*n^2+60*n+36)*x^2+(8*n^2+16*n+8)*x^3)*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[((27 + (54*n + 54)*x + (12*n^2 + 60*n + 36)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(9 + (18*n - 6)*x + 4*n^2*x^2)]];
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] = ((27 + (54*n + 54)*x + (12*n^2 + 60*n + 36)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(9 + (18*n - 6)*x + 4*n^2*x^2)); 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-635f90021d26fb2c
  • 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-635f90021d26fb2c
  • 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-635f90021d26fb2c
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-635f90021d26fb2c

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. [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.