Real-rooted atlas

A034870 → Veronese 2 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 - 3\,n + n x - 3\,n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(3 + 18\,x + 3\,x^{2} - 3\,n - 17\,n x - 8\,n x^{2} - 3\,n^{2} x - n^{2} x^{2} + 2\,n^{3} x + 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-6\,x + 4\,x^{2} + 2\,x^{3} + 4\,n x^{2} - 4\,n x^{3} + 4\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
2112316
313045480
41562101125384
519063084022561792
6113214853696247539678192
71182300312012150156006637836864

All 15 rows

Technical details
Node id
n-afd7c55f05dfbc74 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5953f24a720dd506

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,   2;
  2: 1,  12,    3;
  3: 1,  30,   45,     4;
  4: 1,  56,  210,   112,      5;
  5: 1,  90,  630,   840,    225,      6;
  6: 1, 132, 1485,  3696,   2475,    396,      7;
  7: 1, 182, 3003, 12012,  15015,   6006,    637,     8;
  8: 1, 240, 5460, 32032,  64350,  48048,  12740,   960,    9;
  9: 1, 306, 9180, 74256, 218790, 262548, 129948, 24480, 1377, 10;
%F Formula
The row polynomials satisfy -(3*n-3-(2*n^3-3*n^2+n)*x)*P(n,x) = -(3*n-3-(2*n^3-3*n^2-17*n+18)*x-(6*n^3-n^2-8*n+3)*x^2)*P(n-1,x) - (6*x-(4*n^2+4*n+4)*x^2+(4*n^2+4*n-2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 - (2*n^3 - 3*n^2 - 17*n + 18)*x - (6*n^3 - n^2 - 8*n + 3)*x^2)*P[n - 1, x] - (6*x - (4*n^2 + 4*n + 4)*x^2 + (4*n^2 + 4*n - 2)*x^3)*D[P[n - 1, x], x])/(-3*n + 3 + (2*n^3 - 3*n^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; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n - 3 - (2*n^3 - 3*n^2 - 17*n + 18)*x - (6*n^3 - n^2 - 8*n + 3)*x^2)*v[i - 1] - (6*x - (4*n^2 + 4*n + 4)*x^2 + (4*n^2 + 4*n - 2)*x^3)*deriv(v[i - 1], x))/(-3*n + 3 + (2*n^3 - 3*n^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

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.