Real-rooted atlas

A132440 → θ+1 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2448
3927945
4161129616240
525375625250251300
636111632402340540367308
74930871474917150686010294942973

All 15 rows

Technical details
Node id
n-e22b258e4f44e4b0 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-4732d0911d5a78c5

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:  0;
  1:  1;
  2:  4,     4;
  3:  9,    27,      9;
  4: 16,   112,     96,     16;
  5: 25,   375,    625,    250,     25;
  6: 36,  1116,   3240,   2340,    540,     36;
  7: 49,  3087,  14749,  17150,   6860,   1029,    49;
  8: 64,  8128,  61824, 108864,  67200,  17024,  1792,   64;
  9: 81, 20655, 245025, 629370, 563031, 214326, 37422, 2916, 81;
%F Formula
The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n+(3*n^2-3*n)*x+(n^2-n)*x^2)*P(n-2,x) - ((3*n^2-3*n)*x+(2*n^2-2*n)*x^2)*d/dx P(n-2,x) - (n^2-n)*x^2*d^2/dx^2 P(n-2,x) + (2*n^2-4*n+(2*n^2-4*n)*x)*P(n-1,x) + (2*n^2-4*n)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (3*n^2 - 3*n)*x + (n^2 - n)*x^2)*P[n - 2, x] - ((3*n^2 - 3*n)*x + (2*n^2 - 2*n)*x^2)*D[P[n - 2, x], x] - (n^2 - n)*x^2*D[P[n - 2, x], {x, 2}] + (2*n^2 - 4*n + (2*n^2 - 4*n)*x)*P[n - 1, x] + (2*n^2 - 4*n)*x*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 4 + 4*x; for(n = 3, N, my(i = n + 1); v[i] = (-(n^2 - n + (3*n^2 - 3*n)*x + (n^2 - n)*x^2)*v[i - 2] - ((3*n^2 - 3*n)*x + (2*n^2 - 2*n)*x^2)*deriv(v[i - 2], x) - (n^2 - n)*x^2*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 4*n + (2*n^2 - 4*n)*x)*v[i - 1] + (2*n^2 - 4*n)*x*deriv(v[i - 1], x))/(n^2 - 3*n + 2)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.