Real-rooted atlas

image under θ+1 → (θ+1)² → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n^{2} - 2\,n^{3} + n^{4}\right)\,P_{n}(x) = \left(-n^{2} - 3\,n^{2} x - n^{2} x^{2} - 2\,n^{3} - 6\,n^{3} x - 2\,n^{3} x^{2} - n^{4} - 3\,n^{4} x - n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-3\,n^{2} x - 2\,n^{2} x^{2} - 6\,n^{3} x - 4\,n^{3} x^{2} - 3\,n^{4} x - 2\,n^{4} x^{2}\right)\,P_{n-2}'(x) + \left(-n^{2} x^{2} - 2\,n^{3} x^{2} - n^{4} x^{2}\right)\,P_{n-2}''(x) + \left(2 + 2\,x - 4\,n^{2} - 4\,n^{2} x + 2\,n^{4} + 2\,n^{4} x\right)\,P_{n-1}(x) + \left(2\,x - 4\,n^{2} x + 2\,n^{4} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18816
2278127135
36444838464960
41251875312512501256500
521666961944014040324021643848
634321609103243120050480207203343300811
751265024494592870912537600136192143365122119680

All 15 rows

Technical details
Node id
n-be9c767cabbb35fa · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-4732d0911d5a78c5

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:    8,      8;
  2:   27,     81,      27;
  3:   64,    448,     384,       64;
  4:  125,   1875,    3125,     1250,      125;
  5:  216,   6696,   19440,    14040,     3240,      216;
  6:  343,  21609,  103243,   120050,    48020,     7203,     343;
  7:  512,  65024,  494592,   870912,   537600,   136192,   14336,    512;
  8:  729, 185895, 2205225,  5664330,  5067279,  1928934,  336798,  26244,   729;
  9: 1000, 511000, 9330000, 34105000, 42525000, 22827000, 5880000, 750000, 45000, 1000;
%F Formula
The row polynomials satisfy (n^4-2*n^3+n^2)*P(n,x) = -(n^4+2*n^3+n^2+(3*n^4+6*n^3+3*n^2)*x+(n^4+2*n^3+n^2)*x^2)*P(n-2,x) - ((3*n^4+6*n^3+3*n^2)*x+(2*n^4+4*n^3+2*n^2)*x^2)*d/dx P(n-2,x) - (n^4+2*n^3+n^2)*x^2*d^2/dx^2 P(n-2,x) + (2*n^4-4*n^2+2+(2*n^4-4*n^2+2)*x)*P(n-1,x) + (2*n^4-4*n^2+2)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 + 2*n^3 + n^2 + (3*n^4 + 6*n^3 + 3*n^2)*x + (n^4 + 2*n^3 + n^2)*x^2)*P[n - 2, x] - ((3*n^4 + 6*n^3 + 3*n^2)*x + (2*n^4 + 4*n^3 + 2*n^2)*x^2)*D[P[n - 2, x], x] - (n^4 + 2*n^3 + n^2)*x^2*D[P[n - 2, x], {x, 2}] + (2*n^4 - 4*n^2 + 2 + (2*n^4 - 4*n^2 + 2)*x)*P[n - 1, x] + (2*n^4 - 4*n^2 + 2)*x*D[P[n - 1, x], x])/(n^4 - 2*n^3 + n^2)]];
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] = 8 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 + 2*n^3 + n^2 + (3*n^4 + 6*n^3 + 3*n^2)*x + (n^4 + 2*n^3 + n^2)*x^2)*v[i - 2] - ((3*n^4 + 6*n^3 + 3*n^2)*x + (2*n^4 + 4*n^3 + 2*n^2)*x^2)*deriv(v[i - 2], x) - (n^4 + 2*n^3 + n^2)*x^2*deriv(deriv(v[i - 2], x), x) + (2*n^4 - 4*n^2 + 2 + (2*n^4 - 4*n^2 + 2)*x)*v[i - 1] + (2*n^4 - 4*n^2 + 2)*x*deriv(v[i - 1], x))/(n^4 - 2*n^3 + n^2)); 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

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.