Real-rooted atlas

A038208 → Veronese 2 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 6\,n - n x + n^{3} x\right)\,P_{n}(x) = \left(12 + 24\,x - 4\,x^{2} - 12\,n - 26\,n x - 2\,n x^{2} + 4\,n^{2} x^{2} + 2\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-24\,x + 32\,x^{2} - 8\,x^{3} + 12\,n x^{2} - 12\,n x^{3} + 4\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
24812
384856
41619248256
5326404801152
664192028802565120
7128537613440358422528
8256143365376028672128098304
95123686419353617203223040425984
1010249216064512086016023040061441835008
1120482252802027520378470416896001351687864320
1240965406726082560151388161013760016220162867233554432
138192127795217571840562298885271552014057472745472142606336
14163842981888492011521968046082460057609840230410436608131072603979776
Technical details
Node id
n-47bdf886a6edba7c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9ad63b3855519f4d

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:   2;
  2:   4,     8;
  3:   8,    48;
  4:  16,   192,     48;
  5:  32,   640,    480;
  6:  64,  1920,   2880,    256;
  7: 128,  5376,  13440,   3584;
  8: 256, 14336,  53760,  28672,  1280;
  9: 512, 36864, 193536, 172032, 23040;
%F Formula
The row polynomials satisfy -(6*n-6-(n^3-n)*x)*P(n,x) = -(12*n-12-(2*n^3-26*n+24)*x-(2*n^3+4*n^2-2*n-4)*x^2)*P(n-1,x) - (24*x-(4*n^2+12*n+32)*x^2+(4*n^2+12*n+8)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 12 - (2*n^3 - 26*n + 24)*x - (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*P[n - 1, x] - (24*x - (4*n^2 + 12*n + 32)*x^2 + (4*n^2 + 12*n + 8)*x^3)*D[P[n - 1, x], x])/(-6*n + 6 + (n^3 - 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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n - 12 - (2*n^3 - 26*n + 24)*x - (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*v[i - 1] - (24*x - (4*n^2 + 12*n + 32)*x^2 + (4*n^2 + 12*n + 8)*x^3)*deriv(v[i - 1], x))/(-6*n + 6 + (n^3 - 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

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.