Real-rooted atlas

A347056 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17411
232461088
312033017020640
440018951745470354545
51232954114035651010855632459
6358444044975246944019460221284236348
799841912206147966316802650204981241161201766748

All 15 rows

Technical details
Node id
n-41f0fde874a4c8b3 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-7bc5a9edfc7a801c

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:     7,       4;
  2:    32,      46,       10;
  3:   120,     330,      170,       20;
  4:   400,    1895,     1745,      470,       35;
  5:  1232,    9541,    14035,     6510,     1085,       56;
  6:  3584,   44044,    97524,    69440,    19460,     2212,      84;
  7:  9984,  191220,   614796,   631680,   265020,    49812,    4116,    120;
  8: 26880,  793245,  3616995,  5161590,  3044265,   841890,  113610,   7140,   165;
  9: 70400, 3177295, 20216625, 39055390, 31135335, 11889570, 2330790, 236940, 11715, 220;
%F Formula
The row polynomials satisfy n*P(n,x) = -(2*n+2+(n+1)*x)*P(n-2,x) - (n+1)*x*d/dx P(n-2,x) + (3*n+4+(n+3)*x)*P(n-1,x) + (n+3)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 2 + (n + 1)*x)*P[n - 2, x] - (n + 1)*x*D[P[n - 2, x], x] + (3*n + 4 + (n + 3)*x)*P[n - 1, x] + (n + 3)*x*D[P[n - 1, x], x])/(n)]];
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] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 2 + (n + 1)*x)*v[i - 2] - (n + 1)*x*deriv(v[i - 2], x) + (3*n + 4 + (n + 3)*x)*v[i - 1] + (n + 3)*x*deriv(v[i - 1], x))/(n)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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.