Real-rooted atlas

A007318 → −D²+(−1+n)D+1+n → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2 - 4\,x - 2\,x^{2} - n - 2\,n x - n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 12\,x + 26\,x^{2} + 24\,x^{3} + 8\,x^{4} - n - 6\,n x - 13\,n x^{2} - 12\,n x^{3} - 4\,n x^{4} - n^{2} - 6\,n^{2} x - 13\,n^{2} x^{2} - 12\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-2}(x) + \left(-6 - 24\,x - 30\,x^{2} - 12\,x^{3} + 2\,n^{2} + 8\,n^{2} x + 10\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)
\(P_{2}(x) = 3 + 14\,x + 14\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
23141431
34306644144
4552180256128621
56803808408803522538
671146902080336027849289963
78154113443409520120968288236837908

All 15 rows

Technical details
Node id
n-2d135e27a799111c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-92da879c84c8c477

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:  2,   4;
  2:  3,  14,   14;
  3:  4,  30,   66,    44;
  4:  5,  52,  180,   256,   128;
  5:  6,  80,  380,   840,   880,    352;
  6:  7, 114,  690,  2080,  3360,   2784,    928;
  7:  8, 154, 1134,  4340,  9520,  12096,   8288,   2368;
  8:  9, 200, 1736,  8064, 22400,  38528,  40320,  23552,  5888;
  9: 10, 252, 2520, 13776, 46368, 100800, 142464, 126720, 64512, 14336;
%F Formula
The row polynomials satisfy (n^2-n-2+(2*n^2-2*n-4)*x+(n^2-n-2)*x^2)*P(n,x) = -(n^2+n-2+(6*n^2+6*n-12)*x+(13*n^2+13*n-26)*x^2+(12*n^2+12*n-24)*x^3+(4*n^2+4*n-8)*x^4)*P(n-2,x) + (2*n^2-6+(8*n^2-24)*x+(10*n^2-30)*x^2+(4*n^2-12)*x^3)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 3 + 14*x + 14*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 3 + 14*x + 14*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + n - 2 + (6*n^2 + 6*n - 12)*x + (13*n^2 + 13*n - 26)*x^2 + (12*n^2 + 12*n - 24)*x^3 + (4*n^2 + 4*n - 8)*x^4)*P[n - 2, x] + (2*n^2 - 6 + (8*n^2 - 24)*x + (10*n^2 - 30)*x^2 + (4*n^2 - 12)*x^3)*P[n - 1, x])/(n^2 - n - 2 + (2*n^2 - 2*n - 4)*x + (n^2 - n - 2)*x^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] = 2 + 4*x; v[3] = 3 + 14*x + 14*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(n^2 + n - 2 + (6*n^2 + 6*n - 12)*x + (13*n^2 + 13*n - 26)*x^2 + (12*n^2 + 12*n - 24)*x^3 + (4*n^2 + 4*n - 8)*x^4)*v[i - 2] + (2*n^2 - 6 + (8*n^2 - 24)*x + (10*n^2 - 30)*x^2 + (4*n^2 - 12)*x^3)*v[i - 1])/(n^2 - n - 2 + (2*n^2 - 2*n - 4)*x + (n^2 - n - 2)*x^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

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.