Real-rooted atlas

A048993 → θ+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
122
2325
349215
452421252
5650100452203
6790325360932877
781478401750120418924140
892241862630085053864381221147
9103243696185224170638850121007652115975
10114506750470401597892551501705253732015332678570
11126051155010692051189612564091480380728750114004306924213597
121379218755222750143124350191689265564827640030575053461046141227644437
1314101429172432575359502017096508457233926524918445051006126626501046500122852190899322
14151274437717927928280272513513001882108803946342404440546112404502105195872531558802457321382958545
Technical details
Node id
n-4daa54fde6e196a3 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8452b60480f5fd7f

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:  3,   2;
  3:  4,   9,    2;
  4:  5,  24,   21,     2;
  5:  6,  50,  100,    45,     2;
  6:  7,  90,  325,   360,    93,     2;
  7:  8, 147,  840,  1750,  1204,   189,     2;
  8:  9, 224, 1862,  6300,  8505,  3864,   381,   2;
  9: 10, 324, 3696, 18522, 41706, 38850, 12100, 765, 2;
%F Formula
The row polynomials satisfy P(n,x) = -(1+(n-4)*x-(n-2)*x^2)*P(n-2,x) + (x^2-x^3)*d/dx P(n-2,x) + (2+(n-2)*x)*P(n-1,x) - x^2*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[-(1 + (n - 4)*x - (n - 2)*x^2)*P[n - 2, x] + (x^2 - x^3)*D[P[n - 2, x], x] + (2 + (n - 2)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], 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] = -(1 + (n - 4)*x - (n - 2)*x^2)*v[i - 2] + (x^2 - x^3)*deriv(v[i - 2], x) + (2 + (n - 2)*x)*v[i - 1] - x^2*deriv(v[i - 1], 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.