Real-rooted atlas

A013620 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-4 + 9\,x + 9\,n\right)\,P_{n}(x) = \left(25 + 90\,x + 81\,x^{2} + 45\,n + 81\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
114923
211525281448
3800310534027298036
451252880055890408246561137200
531250230625648000838350459270590492266544
6184375168750062268751166400011317725496011653144136572032
710625001161562553156250130764375183708000142603335520812184782969579774272

All 15 rows

Technical details
Node id
n-9fc2b3f4e7fa1d35 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5e0e612a95dd65b2

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:       14,         9;
  2:      115,       252,         81;
  3:      800,      3105,       3402,         729;
  4:     5125,     28800,      55890,       40824,        6561;
  5:    31250,    230625,     648000,      838350,      459270,       59049;
  6:   184375,   1687500,    6226875,    11664000,    11317725,     4960116,      531441;
  7:  1062500,  11615625,   53156250,   130764375,   183708000,   142603335,    52081218,     4782969;
  8:  6015625,  76500000,  418162500,  1275750000,  2353758750,  2645395200,  1711240020,   535692528,   43046721;
  9: 33593750, 487265625, 3098250000, 11290387500, 25833937500, 38130891750, 35712835200, 19801491660, 5423886846, 387420489;
%F Formula
The row polynomials satisfy (9*n-4+9*x)*P(n,x) = (45*n+25+(81*n+90)*x+81*x^2)*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[((45*n + 25 + (81*n + 90)*x + 81*x^2)*P[n - 1, x])/(9*n - 4 + 9*x)]];
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] = ((45*n + 25 + (81*n + 90)*x + 81*x^2)*v[i - 1])/(9*n - 4 + 9*x)); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.