Real-rooted atlas

A027465 → (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
\(\left(9 - 3\,x + 9\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(27 + 27\,x + 9\,x^{2} + x^{3} + 27\,n x + 15\,n x^{2} + 2\,n x^{3} + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
2924942
3271088116232
481432486192251216
5243162024301440375366144
6729583210935864033756484930208
72187204124592745360236256804102964145408
86561699841837082177281417505443212348153681688128
9196832361967085889797767654503674161111322073621871003211264
1059049787320265720541990403827250220449683349020736032805300012114811136
11177147259815697430851732104018042750121247285501034171072036085549500399314467633152
1253144185030563507510669284160811923756235574433006204123171843247695594000718745184169306184192
131594323276349321243571942702082243518336253039842521838917088006169625332021579150093436210108865911961375731712
14478296989282088435250179103170412814777012251418593176965431467480370176177324147486486009810801141523213841182322256140461056

Computed in your browser ?

Technical details
Node id
n-fae37939907544d7 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1e916cb7d7a3299c

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,      4;
  2:     9,     24,      9;
  3:    27,    108,     81,     16;
  4:    81,    432,    486,    192,     25;
  5:   243,   1620,   2430,   1440,    375,     36;
  6:   729,   5832,  10935,   8640,   3375,    648,     49;
  7:  2187,  20412,  45927,  45360,  23625,   6804,   1029,    64;
  8:  6561,  69984, 183708, 217728, 141750,  54432,  12348,  1536,   81;
  9: 19683, 236196, 708588, 979776, 765450, 367416, 111132, 20736, 2187, 100;
%F Formula
The row polynomials satisfy (9+(9*n-3)*x+n^2*x^2)*P(n,x) = (27+(27*n+27)*x+(3*n^2+15*n+9)*x^2+(n^2+2*n+1)*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 + (27*n + 27)*x + (3*n^2 + 15*n + 9)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(9 + (9*n - 3)*x + 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 + (27*n + 27)*x + (3*n^2 + 15*n + 9)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(9 + (9*n - 3)*x + 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-32b8bc759b20a254
  • 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-32b8bc759b20a254
  • 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-32b8bc759b20a254
  • 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-32b8bc759b20a254

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.