Real-rooted atlas

A137597 → reverse → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 5\,x + 7\,x^{2} + x^{3} - 4\,x^{4} - 2\,x^{5} + n x + 5\,n x^{2} + 9\,n x^{3} + 7\,n x^{4} + 2\,n x^{5}\right)\,P_{n-1}(x) + \left(-x^{2} - 5\,x^{3} - 9\,x^{4} - 7\,x^{5} - 2\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1134
2171119
31124347103
4118109267227622
5125225920173112154117
61334102470768111819710729521
7142686567025781650448510744959227290

All 15 rows

Technical details
Node id
n-6c46082d3ce6c4b2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a0d518f0d2fc546e

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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: 1,  3;
  2: 1,  7,   11;
  3: 1, 12,   43,    47;
  4: 1, 18,  109,   267,    227;
  5: 1, 25,  225,   920,   1731,   1215;
  6: 1, 33,  410,  2470,   7681,  11819,    7107;
  7: 1, 42,  686,  5670,  25781,  65044,   85107,   44959;
  8: 1, 52, 1078, 11662,  72331, 265090,  565013,  645563,  305091;
  9: 1, 63, 1614, 22092, 178563, 888027, 2734201, 5061168, 5147395, 2206399;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1+(n+5)*x+(5*n+7)*x^2+(9*n+1)*x^3+(7*n-4)*x^4+(2*n-2)*x^5)*P(n-1,x) - (x^2+5*x^3+9*x^4+7*x^5+2*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (n + 5)*x + (5*n + 7)*x^2 + (9*n + 1)*x^3 + (7*n - 4)*x^4 + (2*n - 2)*x^5)*P[n - 1, x] - (x^2 + 5*x^3 + 9*x^4 + 7*x^5 + 2*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (n + 5)*x + (5*n + 7)*x^2 + (9*n + 1)*x^3 + (7*n - 4)*x^4 + (2*n - 2)*x^5)*v[i - 1] - (x^2 + 5*x^3 + 9*x^4 + 7*x^5 + 2*x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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

Candidate equalities ?

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.