Real-rooted atlas

A106246 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(-3 - 6\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(3 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 3\,x\)
\(P_{2}(x) = 1 + 6\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1134
216613
319181038
4112364015104
5115601007521272
61189020022512628688
7121126350525441196361696

All 15 rows

Technical details
Node id
n-290229207924eae2 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-5dbc84ad685f9f2d

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: 1,  3;
  2: 1,  6,   6;
  3: 1,  9,  18,  10;
  4: 1, 12,  36,  40,   15;
  5: 1, 15,  60, 100,   75,   21;
  6: 1, 18,  90, 200,  225,  126,   28;
  7: 1, 21, 126, 350,  525,  441,  196,   36;
  8: 1, 24, 168, 560, 1050, 1176,  784,  288,  45;
  9: 1, 27, 216, 840, 1890, 2646, 2352, 1296, 405, 55;
%F Formula
The row polynomials satisfy P(n,x) = (1+3*x+3*x^2+x^3)*P(n-3,x) - (3+6*x+3*x^2)*P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 3*x, P(2,x) = 1 + 6*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[2, x_] := 1 + 6*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x + 3*x^2 + x^3)*P[n - 3, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (3 + 3*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] = 1 + 3*x; v[3] = 1 + 6*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (1 + 3*x + 3*x^2 + x^3)*v[i - 3] - (3 + 6*x + 3*x^2)*v[i - 2] + (3 + 3*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.