Real-rooted atlas

A094305 → basis A013610

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(8 + 36\,x + 54\,x^{2} + 27\,x^{3}\right)\,P_{n-3}(x) + \left(-12 - 36\,x - 27\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 9\,x\)
\(P_{2}(x) = 24 + 72\,x + 54\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16915
2247254150
3803605402701250
424014403240324012159375
56725040151202268017010510365625
6179216128604801209601360808164820412437500
7460848384217728544320816480734832367416787322812500

All 15 rows

Technical details
Node id
n-14200c6b33294c13 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f06e51c139cfae05

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     6,      9;
  2:    24,     72,      54;
  3:    80,    360,     540,     270;
  4:   240,   1440,    3240,    3240,     1215;
  5:   672,   5040,   15120,   22680,    17010,     5103;
  6:  1792,  16128,   60480,  120960,   136080,    81648,    20412;
  7:  4608,  48384,  217728,  544320,   816480,   734832,   367416,    78732;
  8: 11520, 138240,  725760, 2177280,  4082400,  4898880,  3674160,  1574640,  295245;
  9: 28160, 380160, 2280960, 7983360, 17962560, 26943840, 26943840, 17321040, 6495390, 1082565;
%F Formula
The row polynomials satisfy P(n,x) = (8+36*x+54*x^2+27*x^3)*P(n-3,x) - (12+36*x+27*x^2)*P(n-2,x) + (6+9*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 9*x, P(2,x) = 24 + 72*x + 54*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 9*x;
P[2, x_] := 24 + 72*x + 54*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 36*x + 54*x^2 + 27*x^3)*P[n - 3, x] - (12 + 36*x + 27*x^2)*P[n - 2, x] + (6 + 9*x)*P[n - 1, x]];
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] = 6 + 9*x; v[3] = 24 + 72*x + 54*x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 36*x + 54*x^2 + 27*x^3)*v[i - 3] - (12 + 36*x + 27*x^2)*v[i - 2] + (6 + 9*x)*v[i - 1]); 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

Candidate equalities ?

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.