Real-rooted atlas

A038220 → Veronese 2 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(81 - 81\,n - 6\,n x + 6\,n^{3} x\right)\,P_{n}(x) = \left(243 + 216\,x - 16\,x^{2} - 243\,n - 234\,n x - 8\,n x^{2} + 16\,n^{2} x^{2} + 18\,n^{3} x + 8\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-486\,x + 288\,x^{2} - 32\,x^{3} + 108\,n x^{2} - 48\,n x^{3} + 36\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
133
29817
3277299
48143248561
524321607203123
67299720648025617185
721874082445360537693747
86561163296272160645121280507809
9196836298561469664580608345602734371
1059049236196073483204354560518400614414648433
1117714786605203464208028740096570240020275278124995
125314413117787215588936017244057651321600364953628672415039057
1315943231105397286755205609607403524003084804744396811182082197265619
1447829693868890482837186352504388684828021593604981616642348236813107211596679681
Technical details
Node id
n-bab9048ee7ce08fb · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-95af5ce01a82e7fe

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:     3;
  2:     9,      8;
  3:    27,     72;
  4:    81,    432,      48;
  5:   243,   2160,     720;
  6:   729,   9720,    6480,    256;
  7:  2187,  40824,   45360,   5376;
  8:  6561, 163296,  272160,  64512,  1280;
  9: 19683, 629856, 1469664, 580608, 34560;
%F Formula
The row polynomials satisfy -(81*n-81-(6*n^3-6*n)*x)*P(n,x) = -(243*n-243-(18*n^3-234*n+216)*x-(8*n^3+16*n^2-8*n-16)*x^2)*P(n-1,x) - (486*x-(36*n^2+108*n+288)*x^2+(16*n^2+48*n+32)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(243*n - 243 - (18*n^3 - 234*n + 216)*x - (8*n^3 + 16*n^2 - 8*n - 16)*x^2)*P[n - 1, x] - (486*x - (36*n^2 + 108*n + 288)*x^2 + (16*n^2 + 48*n + 32)*x^3)*D[P[n - 1, x], x])/(-81*n + 81 + (6*n^3 - 6*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(243*n - 243 - (18*n^3 - 234*n + 216)*x - (8*n^3 + 16*n^2 - 8*n - 16)*x^2)*v[i - 1] - (486*x - (36*n^2 + 108*n + 288)*x^2 + (16*n^2 + 48*n + 32)*x^3)*deriv(v[i - 1], x))/(-81*n + 81 + (6*n^3 - 6*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

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.