Real-rooted atlas

A126351 → reverse → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6 - 4\,x - x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - x^{2}\right)\,P_{n-2}'(x) + \left(5 + 2\,x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2527
31918340
465110424221
52115703758051241
66652702273098013567218
72059121381770395202170210743807
8630552670106722807242709042843088278111
91917122329061261562664029053566780777043291847242
1058025931502339801045766602817045882882147840132005851012825759
111750993842058183873033199064025440140104435102360589300960212857701192962365
1252734515718430976996022164602442180692801140762483342939657074165727153289099012702294568
1315861316392841051207565514290276001797060265117429312043395352095279184127207081031030490491248135521005933
144766585258885902265627089092569331401436570635511550817278527559032014487387202471168702650648017717707098015471445093176851
Technical details
Node id
n-fe5dc31749d46fb7 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-306ed4af19c47352

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:    19,     18,      3;
  4:    65,    110,     42,      4;
  5:   211,    570,    375,     80,      5;
  6:   665,   2702,   2730,    980,    135,     6;
  7:  2059,  12138,  17703,   9520,   2170,   210,    7;
  8:  6305,  52670, 106722,  80724,  27090,  4284,  308,   8;
  9: 19171, 223290, 612615, 626640, 290535, 66780, 7770, 432, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(6+4*x+x^2)*P(n-2,x) - (2*x+x^2)*d/dx P(n-2,x) + (5+2*x)*P(n-1,x) + 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[-(6 + 4*x + x^2)*P[n - 2, x] - (2*x + x^2)*D[P[n - 2, x], x] + (5 + 2*x)*P[n - 1, x] + 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] = -(6 + 4*x + x^2)*v[i - 2] - (2*x + x^2)*deriv(v[i - 2], x) + (5 + 2*x)*v[i - 1] + 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

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.