Real-rooted atlas

A038208 → Veronese 2 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
244
32424
49632128
5320320640
696019201923072
726888960268814336
871683584021504102465536
91843212902412902418432294912
104608043008064512018432051201310720
111126401351680283852813516801126405767168
12270336405504011354112811008013516802457625165824
1363897611714560421724164217241611714560638976109051904
14149094432800768147603456196804608820019208945664114688469762048
Technical details
Node id
n-ffa0709ca25ec776 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-937db0732045a112

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
  • 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:     0;
  1:     0;
  2:     4;
  3:    24;
  4:    96,     32;
  5:   320,    320;
  6:   960,   1920,    192;
  7:  2688,   8960,   2688;
  8:  7168,  35840,  21504,  1024;
  9: 18432, 129024, 129024, 18432;
%F Formula
The row polynomials satisfy (n^2-3*n+2)*P(n,x) = (2*n^2-2*n+(2*n^2-6*n)*x)*P(n-1,x) + (4*n*x-4*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 2*n + (2*n^2 - 6*n)*x)*P[n - 1, x] + (4*n*x - 4*n*x^2)*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 4; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 2*n + (2*n^2 - 6*n)*x)*v[i - 1] + (4*n*x - 4*n*x^2)*deriv(v[i - 1], x))/(n^2 - 3*n + 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.