Real-rooted atlas

A201701 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n x - 16\,n^{2} - 4\,n^{2} x - n^{2} x^{2} + 16\,n^{3} - 12\,n^{3} x - n^{3} x^{2} + 12\,n^{4} x + n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-16\,x^{2} - 4\,x^{3} + 8\,n x - 8\,n x^{3} - 32\,n^{2} - 72\,n^{2} x - 22\,n^{2} x^{2} - n^{2} x^{3} + 32\,n^{3} + 40\,n^{3} x + 14\,n^{3} x^{2} + 7\,n^{3} x^{3} + 24\,n^{4} x + 22\,n^{4} x^{2} + 5\,n^{4} x^{3} + 2\,n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-16\,x^{2} - 24\,x^{3} - 8\,x^{4} - 24\,n x^{2} - 48\,n x^{3} - 24\,n x^{4} - 32\,n^{2} x - 80\,n^{2} x^{2} - 74\,n^{2} x^{3} - 26\,n^{2} x^{4} - 24\,n^{3} x^{2} - 36\,n^{3} x^{3} - 12\,n^{3} x^{4} - 2\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2246
341216
4832949
5168045141
63219216216402
7644485041121128

All 15 rows

Technical details
Node id
n-be3e6b3dc0d20d55 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c50965050e85f676

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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,    4;
  3:   4,   12;
  4:   8,   32,    9;
  5:  16,   80,   45;
  6:  32,  192,  162,   16;
  7:  64,  448,  504,  112;
  8: 128, 1024, 1440,  512,  25;
  9: 256, 2304, 3888, 1920, 225;
%F Formula
The row polynomials satisfy (16*n^3-16*n^2+(12*n^4-12*n^3-4*n^2+4*n)*x+(n^5+n^4-n^3-n^2)*x^2)*P(n,x) = (32*n^3-32*n^2+(24*n^4+40*n^3-72*n^2+8*n)*x+(2*n^5+22*n^4+14*n^3-22*n^2-16)*x^2+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^3)*P(n-1,x) - (32*n^2*x+(24*n^3+80*n^2+24*n+16)*x^2+(2*n^4+36*n^3+74*n^2+48*n+24)*x^3+(2*n^4+12*n^3+26*n^2+24*n+8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((32*n^3 - 32*n^2 + (24*n^4 + 40*n^3 - 72*n^2 + 8*n)*x + (2*n^5 + 22*n^4 + 14*n^3 - 22*n^2 - 16)*x^2 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^3)*P[n - 1, x] - (32*n^2*x + (24*n^3 + 80*n^2 + 24*n + 16)*x^2 + (2*n^4 + 36*n^3 + 74*n^2 + 48*n + 24)*x^3 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)*D[P[n - 1, x], x])/(16*n^3 - 16*n^2 + (12*n^4 - 12*n^3 - 4*n^2 + 4*n)*x + (n^5 + n^4 - n^3 - n^2)*x^2)]];
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; for(n = 2, N, my(i = n + 1); v[i] = ((32*n^3 - 32*n^2 + (24*n^4 + 40*n^3 - 72*n^2 + 8*n)*x + (2*n^5 + 22*n^4 + 14*n^3 - 22*n^2 - 16)*x^2 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^3)*v[i - 1] - (32*n^2*x + (24*n^3 + 80*n^2 + 24*n + 16)*x^2 + (2*n^4 + 36*n^3 + 74*n^2 + 48*n + 24)*x^3 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)*deriv(v[i - 1], x))/(16*n^3 - 16*n^2 + (12*n^4 - 12*n^3 - 4*n^2 + 4*n)*x + (n^5 + n^4 - n^3 - n^2)*x^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

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.