Real-rooted atlas

A027467 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(14 + x + n\right)\,P_{n}(x) = \left(225 + 30\,x + x^{2} + 15\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
116117
2255321288
340507654814864
46412516200153064181920
510125003206254050025508011376256
615946875607500096187581000382596123068672
725059375011162812521262500224437514175053551121385875968

All 15 rows

Technical details
Node id
n-3968007280518c14 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-214d56e6a3c38c2f

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:          16,           1;
  2:         255,          32,          1;
  3:        4050,         765,         48,          1;
  4:       64125,       16200,       1530,         64,         1;
  5:     1012500,      320625,      40500,       2550,        80,       1;
  6:    15946875,     6075000,     961875,      81000,      3825,      96,      1;
  7:   250593750,   111628125,   21262500,    2244375,    141750,    5355,    112,    1;
  8:  3929765625,  2004750000,  446512500,   56700000,   4488750,  226800,   7140,  128,   1;
  9: 61509375000, 35367890625, 9021375000, 1339537500, 127575000, 8079750, 340200, 9180, 144, 1;
%F Formula
The row polynomials satisfy (n+14+x)*P(n,x) = (15*n+225+(n+30)*x+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[((15*n + 225 + (n + 30)*x + x^2)*P[n - 1, x])/(n + 14 + 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] = ((15*n + 225 + (n + 30)*x + x^2)*v[i - 1])/(n + 14 + 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.