Real-rooted atlas

A216973 → θ+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
244
3121830
4247248144
540180240100560
6603607206001801920
7846301680210012602946048
81121008336056005040235244817920
914415126048126001512010584403264850688
10180216010080252003780035280201606480900138240
11220297015840462008316097020739203564099001210366080
122643960237607920016632023284822176014256059400145201584946176
13312514834320128700308880504504576576463320257400943802059220282396160
143646552480482002005405401009008134534412972969009004404401441442839225485963776
Technical details
Node id
n-82c964c6f2d99674 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-72ae8e6419635e63

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (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): yes

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:  12,   18;
  4:  24,   72,   48;
  5:  40,  180,  240,   100;
  6:  60,  360,  720,   600,   180;
  7:  84,  630, 1680,  2100,  1260,   294;
  8: 112, 1008, 3360,  5600,  5040,  2352,  448;
  9: 144, 1512, 6048, 12600, 15120, 10584, 4032, 648;
%F Formula
The row polynomials satisfy (2*n-4+(n^2-3*n+2)*x)*P(n,x) = (2*n+(n^2+2*n)*x+n^2*x^2)*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 + (n^2 + 2*n)*x + n^2*x^2)*P[n - 1, x])/(2*n - 4 + (n^2 - 3*n + 2)*x)]];
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 + (n^2 + 2*n)*x + n^2*x^2)*v[i - 1])/(2*n - 4 + (n^2 - 3*n + 2)*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.