Real-rooted atlas

A038231 → (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
\(\left(16 - 4\,x + 12\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(64 + 48\,x + 12\,x^{2} + x^{3} + 48\,n x + 20\,n x^{2} + 2\,n x^{3} + 4\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
21632957
36419210816380
42561024864256252425
510245120576025605003615000
6409624576345602048060008644990625
7163841146881935361433605600012096137264537500
8655365242881032192917504448000129024219522048813140625
92621442359296530841655050243225600116121626342436864291610018125000
1010485761048576026542080314572802150400092897282634240491520583204000121103515625
1141943044613734412976128017301504013516800068124672231813125406720855360880005324144585937500
12167772162013265926228541449227468808110080004671406081854504965190451210264320140800012777669121693291015625
136710886487241523229444014084798283776468582400030364139521377632256449839104106748928183040002214784179712878819618359375000
142684354563758096384137405399042442762649626240614400188932423689643425792359871283299632332820500480031006976335462424606410976225101806640625

Computed in your browser ?

Technical details
Node id
n-d861aaa140b3a14a · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9e3a89ee8f65d0f5

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:      4,       4;
  2:     16,      32,       9;
  3:     64,     192,     108,      16;
  4:    256,    1024,     864,     256,      25;
  5:   1024,    5120,    5760,    2560,     500,      36;
  6:   4096,   24576,   34560,   20480,    6000,     864,     49;
  7:  16384,  114688,  193536,  143360,   56000,   12096,   1372,    64;
  8:  65536,  524288, 1032192,  917504,  448000,  129024,  21952,  2048,   81;
  9: 262144, 2359296, 5308416, 5505024, 3225600, 1161216, 263424, 36864, 2916, 100;
%F Formula
The row polynomials satisfy (16+(12*n-4)*x+n^2*x^2)*P(n,x) = (64+(48*n+48)*x+(4*n^2+20*n+12)*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[((64 + (48*n + 48)*x + (4*n^2 + 20*n + 12)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(16 + (12*n - 4)*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] = ((64 + (48*n + 48)*x + (4*n^2 + 20*n + 12)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(16 + (12*n - 4)*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-831f84c152bb1a61
  • 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-831f84c152bb1a61
  • 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-831f84c152bb1a61
  • 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-831f84c152bb1a61

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.