Real-rooted atlas

A007318 → ×(1+2n+n²) → x ↦ x+1

nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18412
23636981
31281929616432
4400800600200252025
51152288028801440360368748
63136940811760784029405884935721
7819228672430083584017920537689664139968
8207368294414515214515290720362889072129681531441
951200230400460800537600403200201600672001440018001001968300
101239046195201393920185856016262409757444065601161602178024201217144929
1129491216220164055040608256060825604257792212889676032019008031680316814425509168
1269222441533441142169619036160214156801713254499939844283136133848029744044616405616989813529
1316056321043660831309824574013447175168064576512430510082152550480720642242240448448611525096196312487308
143686400258048008386560016773120023063040023063040017297280098841600432432001441440036036006552008190063002251076168025

Computed in your browser ?

Technical details
Node id
n-63d7f3b1eb8cf1f2 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-11aec07ce60b6d61

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): yes

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:     8,      4;
  2:    36,     36,      9;
  3:   128,    192,     96,     16;
  4:   400,    800,    600,    200,     25;
  5:  1152,   2880,   2880,   1440,    360,     36;
  6:  3136,   9408,  11760,   7840,   2940,    588,    49;
  7:  8192,  28672,  43008,  35840,  17920,   5376,   896,    64;
  8: 20736,  82944, 145152, 145152,  90720,  36288,  9072,  1296,   81;
  9: 51200, 230400, 460800, 537600, 403200, 201600, 67200, 14400, 1800, 100;
%F Formula
The row polynomials satisfy n^2*P(n,x) = (2*n^2+4*n+2+(n^2+2*n+1)*x)*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[((2*n^2 + 4*n + 2 + (n^2 + 2*n + 1)*x)*P[n - 1, x])/(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; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 4*n + 2 + (n^2 + 2*n + 1)*x)*v[i - 1])/(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-18ace392e620d7bd
  • 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-18ace392e620d7bd
  • 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-18ace392e620d7bd

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 (67)

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.