Real-rooted atlas

A007318 → −D²+(−1+n)D+1+n → x ↦ x+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(-2 - n + n^{2}\right)\,P_{n}(x) = \left(8 + 8\,x + 2\,x^{2} - 4\,n - 4\,n x - n x^{2} - 4\,n^{2} - 4\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-12 - 6\,x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
\(P_{2}(x) = 14 + 14\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21414331
34466304144
4128256180525621
53528808403808062538
692827843360208069011479963
72368828812096952043401134154837908
8588823552403203852822400806417362009140697
914336645121267201424641008004636813776252025210511758
103430417152038016049152040992022982487360220803510310111830519
11808964449281098240160512015417601020096473088153120336604730374126456024
121884161130496307507250124805448960415641622767368997122534404928062044441322497669
13434176282214483865601508249618304000158146569993984466752016061764004006978479565201477590386
14991232693862422364160439828485893888056888832407447042196480089369282722720608608960961001060215265189059

Generated b-files: A094951 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-aff30617856e629d · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-92da879c84c8c477

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 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,     2;
  2:    14,    14,      3;
  3:    44,    66,     30,      4;
  4:   128,   256,    180,     52,      5;
  5:   352,   880,    840,    380,     80,     6;
  6:   928,  2784,   3360,   2080,    690,   114,     7;
  7:  2368,  8288,  12096,   9520,   4340,  1134,   154,    8;
  8:  5888, 23552,  40320,  38528,  22400,  8064,  1736,  200,   9;
  9: 14336, 64512, 126720, 142464, 100800, 46368, 13776, 2520, 252, 10;
%F Formula
The row polynomials satisfy (n^2-n-2)*P(n,x) = -(4*n^2+4*n-8+(4*n^2+4*n-8)*x+(n^2+n-2)*x^2)*P(n-2,x) + (4*n^2-12+(2*n^2-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 14 + 14*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[2, x_] := 14 + 14*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 + 4*n - 8 + (4*n^2 + 4*n - 8)*x + (n^2 + n - 2)*x^2)*P[n - 2, x] + (4*n^2 - 12 + (2*n^2 - 6)*x)*P[n - 1, x])/(n^2 - n - 2)]];
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] = 4 + 2*x; v[3] = 14 + 14*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 + 4*n - 8 + (4*n^2 + 4*n - 8)*x + (n^2 + n - 2)*x^2)*v[i - 2] + (4*n^2 - 12 + (2*n^2 - 6)*x)*v[i - 1])/(n^2 - n - 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, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-2c04e0d67d228006
  • 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-2c04e0d67d228006
  • 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-2c04e0d67d228006

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.