Real-rooted atlas

A147716 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(15 + 2\,n x\right)\,P_{n}(x) = \left(225 + 60\,x + 4\,x^{2} + 30\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
115419
222512012357
333752700540326647
4506255400016200192080122825
575937510125004050007200060001922255067
61139062518225000911250021600002700001728044841175853
7170859375318937500191362500567000009450000907200470401024748264639

All 15 rows

Technical details
Node id
n-de3ba732ba8db835 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6200f7cf98c2da4e

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:          15,           4;
  2:         225,         120,          12;
  3:        3375,        2700,         540,          32;
  4:       50625,       54000,       16200,        1920,         80;
  5:      759375,     1012500,      405000,       72000,       6000,        192;
  6:    11390625,    18225000,     9112500,     2160000,     270000,      17280,       448;
  7:   170859375,   318937500,   191362500,    56700000,    9450000,     907200,     47040,    1024;
  8:  2562890625,  5467500000,  3827250000,  1360800000,  283500000,   36288000,   2822400,  122880,   2304;
  9: 38443359375, 92264062500, 73811250000, 30618000000, 7654500000, 1224720000, 127008000, 8294400, 311040, 5120;
%F Formula
The row polynomials satisfy (15+2*n*x)*P(n,x) = (225+(30*n+60)*x+(4*n+4)*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[((225 + (30*n + 60)*x + (4*n + 4)*x^2)*P[n - 1, x])/(15 + 2*n*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] = ((225 + (30*n + 60)*x + (4*n + 4)*x^2)*v[i - 1])/(15 + 2*n*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.