Real-rooted atlas

A003506 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
24812
312482787
43219221664504
58064010806401252565
6192192043203840150021611988
74485376151201792010500302434352731

All 15 rows

Technical details
Node id
n-3073917454e249df · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:    0;
  1:    1;
  2:    4,     8;
  3:   12,    48,     27;
  4:   32,   192,    216,     64;
  5:   80,   640,   1080,    640,    125;
  6:  192,  1920,   4320,   3840,   1500,    216;
  7:  448,  5376,  15120,  17920,  10500,   3024,   343;
  8: 1024, 14336,  48384,  71680,  56000,  24192,  5488,  512;
  9: 2304, 36864, 145152, 258048, 252000, 145152, 49392, 9216, 729;
%F Formula
The row polynomials satisfy (4*n-4+(6*n^2-14*n+8)*x+(n^3-3*n^2+3*n-1)*x^2)*P(n,x) = (8*n+12*n^2*x+(2*n^3+6*n^2-2*n)*x^2+n^3*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 12*n^2*x + (2*n^3 + 6*n^2 - 2*n)*x^2 + n^3*x^3)*P[n - 1, x])/(4*n - 4 + (6*n^2 - 14*n + 8)*x + (n^3 - 3*n^2 + 3*n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((8*n + 12*n^2*x + (2*n^3 + 6*n^2 - 2*n)*x^2 + n^3*x^3)*v[i - 1])/(4*n - 4 + (6*n^2 - 14*n + 8)*x + (n^3 - 3*n^2 + 3*n - 1)*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

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.