Real-rooted atlas

A013620 → basis A008288 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3\,x - 10\,x^{2}\right)\,P_{n-2}(x) + \left(3 + 7\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1358
29392573
327189333125674
481783234923916256229
52432997128252272515657312557572
672910935609931571671897299693915625532117
721873863726608590695716092811438641578253781254918166

All 15 rows

Technical details
Node id
n-7b22e4ad64916eb8 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-0fee5d4c7376c95e

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     3,      5;
  2:     9,     39,      25;
  3:    27,    189,     333,      125;
  4:    81,    783,    2349,     2391,      625;
  5:   243,   2997,   12825,    22725,    15657,      3125;
  6:   729,  10935,   60993,   157167,   189729,     96939,     15625;
  7:  2187,  38637,  266085,   906957,  1609281,   1438641,    578253,    78125;
  8:  6561, 133407, 1094229,  4657095, 11038113,  14578407,  10198773,  3359631,   390625;
  9: 19683, 452709, 4310577, 22042773, 65774025, 116760285, 120868281, 68818653, 19141137, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (3*x-10*x^2)*P(n-2,x) + (3+7*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 5*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 5*x;
P[n_, x_] := P[n, x] = Expand[(3*x - 10*x^2)*P[n - 2, x] + (3 + 7*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] = 3 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (3*x - 10*x^2)*v[i - 2] + (3 + 7*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

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