Real-rooted atlas

basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.

only rational roots

Proved for all n all roots real, in (−∞, 1]all roots real, in [0, ∞)alternating coefficient signsinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1-330
26-1260
3-1030-30100
415-6090-60150
5-21105-210210-105210
628-168420-560420-168280
7-36252-7561260-1260756-252360

All 15 rows

Technical details
Node id
n-dcce4b7f491234bd · revision 0 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A127773
Fingerprint
f6-85f6fce8b02c5e6d

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): no
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:  -3,    3;
  2:   6,  -12,     6;
  3: -10,   30,   -30,    10;
  4:  15,  -60,    90,   -60,    15;
  5: -21,  105,  -210,   210,  -105,    21;
  6:  28, -168,   420,  -560,   420,  -168,    28;
  7: -36,  252,  -756,  1260, -1260,   756,  -252,   36;
  8:  45, -360,  1260, -2520,  3150, -2520,  1260, -360,   45;
  9: -55,  495, -1980,  4620, -6930,  6930, -4620, 1980, -495, 55;
%F Formula
The row polynomials satisfy -n*P(n,x) = (n+2-(n+2)*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[((n + 2 - (n + 2)*x)*P[n - 1, x])/(-n)]];
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] = ((n + 2 - (n + 2)*x)*v[i - 1])/(-n)); 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)

polytool c48629374597

Properties

Proof chains ? JSON

Used in 1

Same fingerprint ?

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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [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
  6. [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
  7. [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
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.