Real-rooted atlas

A137597 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(5 + 14\,x + 12\,x^{2} + 3\,x^{3} - 2\,n - 7\,n x - 8\,n x^{2} - 3\,n x^{3}\right)\,P_{n-1}(x) + \left(2 + 9\,x + 15\,x^{2} + 11\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
211291959
347184239103573
42271175227219456226241
512157806199942552116234411774887
671075446117338129352427870114072429521977419
7449593998201519825320117140349393043566127212322729013743693

All 15 rows

Technical details
Node id
n-2bfda822914612fb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c4bff110fe4eab08

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:       3,        4;
  2:      11,       29,        19;
  3:      47,      184,       239,       103;
  4:     227,     1175,      2272,      1945,       622;
  5:    1215,     7806,     19994,     25521,     16234,      4117;
  6:    7107,    54461,    173381,    293524,    278701,    140724,     29521;
  7:   44959,   399820,   1519825,   3201171,   4034939,   3043566,   1272123,    227290;
  8:  305091,  3086291,  13626502,  34297087,  53824051,  53931947,  33695516,  12001715,   1865881;
  9: 2206399, 25004986, 125670692, 367626953, 689838155, 861096517, 715033150, 380874505, 118094087, 16239523;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(2*n-5+(7*n-14)*x+(8*n-12)*x^2+(3*n-3)*x^3)*P(n-1,x) + (2+9*x+15*x^2+11*x^3+3*x^4)*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[(-(2*n - 5 + (7*n - 14)*x + (8*n - 12)*x^2 + (3*n - 3)*x^3)*P[n - 1, x] + (2 + 9*x + 15*x^2 + 11*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + x)]];
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] = (-(2*n - 5 + (7*n - 14)*x + (8*n - 12)*x^2 + (3*n - 3)*x^3)*v[i - 1] + (2 + 9*x + 15*x^2 + 11*x^3 + 3*x^4)*deriv(v[i - 1], x))/(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.