Real-rooted atlas

A120910 → Veronese 2 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1152 - 96\,x + 1152\,n + 80\,n x + 192\,n^{2} x - 2\,n^{2} x^{2} - 176\,n^{3} x - 2\,n^{3} x^{2} + 2\,n^{4} x^{2} + 2\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-2304 - 2496\,x + 16\,x^{2} - 4\,x^{3} + 2304\,n + 2464\,n x + 248\,n x^{2} - 8\,n x^{3} + 384\,n^{2} x - 116\,n^{2} x^{2} - n^{2} x^{3} - 352\,n^{3} x - 156\,n^{3} x^{2} + 7\,n^{3} x^{3} + 4\,n^{4} x^{2} + 5\,n^{4} x^{3} + 4\,n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(4608\,x - 1408\,x^{2} + 96\,x^{3} - 8\,x^{4} - 1344\,n x^{2} + 432\,n x^{3} - 24\,n x^{4} - 704\,n^{2} x^{2} + 280\,n^{2} x^{3} - 26\,n^{2} x^{4} + 48\,n^{3} x^{3} - 12\,n^{3} x^{4} + 8\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 3\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3Pn(1)
033
166
2121224
3247296
44828827363
5969602701326
619228801620484740
73848064756067216680

All 15 rows

Technical details
Node id
n-fe249fd2e8b8c5b9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-0a3780febcbb539d

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:    3;
  1:    6;
  2:   12,    12;
  3:   24,    72;
  4:   48,   288,     27;
  5:   96,   960,    270;
  6:  192,  2880,   1620,    48;
  7:  384,  8064,   7560,   672;
  8:  768, 21504,  30240,  5376,   75;
  9: 1536, 55296, 108864, 32256, 1350;
%F Formula
The row polynomials satisfy (1152*n-1152-(176*n^3-192*n^2-80*n+96)*x+(2*n^5+2*n^4-2*n^3-2*n^2)*x^2)*P(n,x) = (2304*n-2304-(352*n^3-384*n^2-2464*n+2496)*x+(4*n^5+4*n^4-156*n^3-116*n^2+248*n+16)*x^2+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^3)*P(n-1,x) + (4608*x-(704*n^2+1344*n+1408)*x^2+(8*n^4+48*n^3+280*n^2+432*n+96)*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) = 3, P(1,x) = 6.
%t Mathematica
P[0, x_] := 3;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((2304*n - 2304 - (352*n^3 - 384*n^2 - 2464*n + 2496)*x + (4*n^5 + 4*n^4 - 156*n^3 - 116*n^2 + 248*n + 16)*x^2 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^3)*P[n - 1, x] + (4608*x - (704*n^2 + 1344*n + 1408)*x^2 + (8*n^4 + 48*n^3 + 280*n^2 + 432*n + 96)*x^3 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)*D[P[n - 1, x], x])/(1152*n - 1152 - (176*n^3 - 192*n^2 - 80*n + 96)*x + (2*n^5 + 2*n^4 - 2*n^3 - 2*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] = 3; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((2304*n - 2304 - (352*n^3 - 384*n^2 - 2464*n + 2496)*x + (4*n^5 + 4*n^4 - 156*n^3 - 116*n^2 + 248*n + 16)*x^2 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^3)*v[i - 1] + (4608*x - (704*n^2 + 1344*n + 1408)*x^2 + (8*n^4 + 48*n^3 + 280*n^2 + 432*n + 96)*x^3 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)*deriv(v[i - 1], x))/(1152*n - 1152 - (176*n^3 - 192*n^2 - 80*n + 96)*x + (2*n^5 + 2*n^4 - 2*n^3 - 2*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.