Real-rooted atlas

A125092 → reverse → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 4\,x - 6\,x^{2} - 4\,x^{3} - x^{4}\right)\,P_{n-4}(x) + \left(4 + 12\,x + 12\,x^{2} + 4\,x^{3}\right)\,P_{n-3}(x) + \left(-6 - 12\,x - 6\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 8 + 2\,x\)
\(P_{3}(x) = 27 + 24\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
28210
32724354
464108484224
5125320270805800
621675096054012062592
734315122625224094516877840
851227446048700044801512224822528
9729460812348181441575080642268288962208
10100072902304041160453603150013440324036010166400
111331110004009584480113190997925775021120445544011433664
121728159726600016038025344027165619958499000316805940528121105920
13219722464103818286000521235658944588588370656160875457607722624132768896
142744307581572484844841001000145945815375361177176648648250250640649828728146823936
Technical details
Node id
n-daef2e79c4033d27 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d42e6339eab21b9c

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:   1;
  2:   8,    2;
  3:  27,   24,     3;
  4:  64,  108,    48,     4;
  5: 125,  320,   270,    80,     5;
  6: 216,  750,   960,   540,   120,    6;
  7: 343, 1512,  2625,  2240,   945,  168,    7;
  8: 512, 2744,  6048,  7000,  4480, 1512,  224,   8;
  9: 729, 4608, 12348, 18144, 15750, 8064, 2268, 288, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(1+4*x+6*x^2+4*x^3+x^4)*P(n-4,x) + (4+12*x+12*x^2+4*x^3)*P(n-3,x) - (6+12*x+6*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 8 + 2*x, P(3,x) = 27 + 24*x + 3*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 8 + 2*x;
P[3, x_] := 27 + 24*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*P[n - 4, x] + (4 + 12*x + 12*x^2 + 4*x^3)*P[n - 3, x] - (6 + 12*x + 6*x^2)*P[n - 2, x] + (4 + 4*x)*P[n - 1, 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] = 1; v[3] = 8 + 2*x; v[4] = 27 + 24*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*v[i - 4] + (4 + 12*x + 12*x^2 + 4*x^3)*v[i - 3] - (6 + 12*x + 6*x^2)*v[i - 2] + (4 + 4*x)*v[i - 1]); 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.