Real-rooted atlas

A027465 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-9 + 12\,x - x^{2} + 9\,n - 21\,n x + 3\,n x^{2} + 9\,n^{2} x - 3\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(27\,n - 3\,n x^{2} + 27\,n^{2} x + 9\,n^{2} x^{2} + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
26814
3277227126
410843232464928
5405216024309601256080
614589720145808640225021636864
75103408247654560480236254536343211456

All 15 rows

Technical details
Node id
n-4b3fd66b22df45fb · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1e916cb7d7a3299c

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:     6,      8;
  3:    27,     72,      27;
  4:   108,    432,     324,      64;
  5:   405,   2160,    2430,     960,     125;
  6:  1458,   9720,   14580,    8640,    2250,    216;
  7:  5103,  40824,   76545,   60480,   23625,   4536,    343;
  8: 17496, 163296,  367416,  362880,  189000,  54432,   8232,   512;
  9: 59049, 629856, 1653372, 1959552, 1275750, 489888, 111132, 13824, 729;
%F Formula
The row polynomials satisfy (9*n-9+(9*n^2-21*n+12)*x+(n^3-3*n^2+3*n-1)*x^2)*P(n,x) = (27*n+27*n^2*x+(3*n^3+9*n^2-3*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[((27*n + 27*n^2*x + (3*n^3 + 9*n^2 - 3*n)*x^2 + n^3*x^3)*P[n - 1, x])/(9*n - 9 + (9*n^2 - 21*n + 12)*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] = ((27*n + 27*n^2*x + (3*n^3 + 9*n^2 - 3*n)*x^2 + n^3*x^3)*v[i - 1])/(9*n - 9 + (9*n^2 - 21*n + 12)*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.