Real-rooted atlas

A013609 → Veronese 2 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-9 - 12\,x + 9\,n + 10\,n x + 24\,n^{2} x - 4\,n^{2} x^{2} - 22\,n^{3} x - 4\,n^{3} x^{2} + 4\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-9 - 156\,x + 16\,x^{2} - 64\,x^{3} + 9\,n + 154\,n x + 248\,n x^{2} - 128\,n x^{3} + 24\,n^{2} x - 116\,n^{2} x^{2} - 16\,n^{2} x^{3} - 22\,n^{3} x - 156\,n^{3} x^{2} + 112\,n^{3} x^{3} + 4\,n^{4} x^{2} + 80\,n^{4} x^{3} + 4\,n^{5} x^{2} + 16\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(18\,x - 88\,x^{2} + 96\,x^{3} - 128\,x^{4} - 84\,n x^{2} + 432\,n x^{3} - 384\,n x^{4} - 44\,n^{2} x^{2} + 280\,n^{2} x^{3} - 416\,n^{2} x^{4} + 48\,n^{3} x^{3} - 192\,n^{3} x^{4} + 8\,n^{4} x^{3} - 32\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
211617
314849
4196144241
51160720881
61240216010243425
713365040716812545

All 15 rows

Technical details
Node id
n-c3bd081c28f27dc0 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-104b93e0fc2039fc

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: 1,  16;
  3: 1,  48;
  4: 1,  96,   144;
  5: 1, 160,   720;
  6: 1, 240,  2160,  1024;
  7: 1, 336,  5040,  7168;
  8: 1, 448, 10080, 28672,  6400;
  9: 1, 576, 18144, 86016, 57600;
%F Formula
The row polynomials satisfy (9*n-9-(22*n^3-24*n^2-10*n+12)*x+(4*n^5+4*n^4-4*n^3-4*n^2)*x^2)*P(n,x) = (9*n-9-(22*n^3-24*n^2-154*n+156)*x+(4*n^5+4*n^4-156*n^3-116*n^2+248*n+16)*x^2+(16*n^5+80*n^4+112*n^3-16*n^2-128*n-64)*x^3)*P(n-1,x) + (18*x-(44*n^2+84*n+88)*x^2+(8*n^4+48*n^3+280*n^2+432*n+96)*x^3-(32*n^4+192*n^3+416*n^2+384*n+128)*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[((9*n - 9 - (22*n^3 - 24*n^2 - 154*n + 156)*x + (4*n^5 + 4*n^4 - 156*n^3 - 116*n^2 + 248*n + 16)*x^2 + (16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64)*x^3)*P[n - 1, x] + (18*x - (44*n^2 + 84*n + 88)*x^2 + (8*n^4 + 48*n^3 + 280*n^2 + 432*n + 96)*x^3 - (32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x^4)*D[P[n - 1, x], x])/(9*n - 9 - (22*n^3 - 24*n^2 - 10*n + 12)*x + (4*n^5 + 4*n^4 - 4*n^3 - 4*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] = ((9*n - 9 - (22*n^3 - 24*n^2 - 154*n + 156)*x + (4*n^5 + 4*n^4 - 156*n^3 - 116*n^2 + 248*n + 16)*x^2 + (16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64)*x^3)*v[i - 1] + (18*x - (44*n^2 + 84*n + 88)*x^2 + (8*n^4 + 48*n^3 + 280*n^2 + 432*n + 96)*x^3 - (32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x^4)*deriv(v[i - 1], x))/(9*n - 9 - (22*n^3 - 24*n^2 - 10*n + 12)*x + (4*n^5 + 4*n^4 - 4*n^3 - 4*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

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.