Real-rooted atlas

A352880 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(3 + 7\,n + 5\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(12 + 26\,n + 10\,n x + 18\,n^{2} + 11\,n^{2} x + 4\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(-6\,x - 6\,x^{2} - 7\,n x - 7\,n x^{2} - 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 8\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
088
1301242
21129616224
3420560210201210
4158428801728384246600
56006138601155042006302836274
622880640646864035200880096032200576
78751628828837837825225290090166321386361114578

All 15 rows

Technical details
Node id
n-0248ad0b5ffb0432 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-63d66b489de6a076

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:       8;
  1:      30,      12;
  2:     112,      96,      16;
  3:     420,     560,     210,      20;
  4:    1584,    2880,    1728,     384,      24;
  5:    6006,   13860,   11550,    4200,     630,      28;
  6:   22880,   64064,   68640,   35200,    8800,     960,     32;
  7:   87516,  288288,  378378,  252252,   90090,   16632,   1386,    36;
  8:  335920, 1272960, 1980160, 1630720,  764400,  203840,  29120,  1920,   40;
  9: 1293292, 5542680, 9976824, 9801792, 5717712, 2018016, 420420, 48048, 2574, 44;
%F Formula
The row polynomials satisfy (n^3+5*n^2+7*n+3)*P(n,x) = (4*n^3+18*n^2+26*n+12+(3*n^3+11*n^2+10*n)*x)*P(n-1,x) - ((2*n^2+7*n+6)*x+(2*n^2+7*n+6)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 8.
%t Mathematica
P[0, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^3 + 18*n^2 + 26*n + 12 + (3*n^3 + 11*n^2 + 10*n)*x)*P[n - 1, x] - ((2*n^2 + 7*n + 6)*x + (2*n^2 + 7*n + 6)*x^2)*D[P[n - 1, x], x])/(n^3 + 5*n^2 + 7*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 8; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^3 + 18*n^2 + 26*n + 12 + (3*n^3 + 11*n^2 + 10*n)*x)*v[i - 1] - ((2*n^2 + 7*n + 6)*x + (2*n^2 + 7*n + 6)*x^2)*deriv(v[i - 1], x))/(n^3 + 5*n^2 + 7*n + 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.