Real-rooted atlas

A049020 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 4\,x - x^{2}\right)\,P_{n-2}(x) + \left(-4 - 4\,x - x^{2}\right)\,P_{n-2}'(x) + \left(5 + 2\,x\right)\,P_{n-1}(x) + \left(2 + x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2527
32518346
4133134424313
57619704358052251
64677714241101100135617170
73074554362378631316023802107138727
821517343039034981814848435210462030881184011
91596217355117832855791639008478275829088274432910641880
1012501189305394623158871018041020622576513364821772401392058510100424383
11102995289273515066311998863200170520794964502028965433492693511202227577011992189287
128898439732548046294317129003422529932041009584840297904200587303647695336653895342109901210236777352
13803927103324657033258331964788352582302465612862377385429949977698386288015453609616465878115687050765124813110033758693
147575886061324749259043835790609199830209913879616531042027561637205630160343543362933229728375801426331931601959958731641547141229582921083
Technical details
Node id
n-8151ac3bf8b01324 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-0bcc5e6a0080be5a

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:       5,       2;
  3:      25,      18,       3;
  4:     133,     134,      42,       4;
  5:     761,     970,     435,      80,      5;
  6:    4677,    7142,    4110,    1100,    135,     6;
  7:   30745,   54362,   37863,   13160,   2380,   210,    7;
  8:  215173,  430390,  349818,  148484,  35210,  4620,  308,   8;
  9: 1596217, 3551178, 3285579, 1639008, 478275, 82908, 8274, 432, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(4+4*x+x^2)*P(n-2,x) - (4+4*x+x^2)*d/dx P(n-2,x) + (5+2*x)*P(n-1,x) + (2+x)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4 + 4*x + x^2)*P[n - 2, x] - (4 + 4*x + x^2)*D[P[n - 2, x], x] + (5 + 2*x)*P[n - 1, x] + (2 + x)*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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(4 + 4*x + x^2)*v[i - 2] - (4 + 4*x + x^2)*deriv(v[i - 2], x) + (5 + 2*x)*v[i - 1] + (2 + x)*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

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.