Real-rooted atlas

A013622 → x ↦ x/(x+1) → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 24\,x + 72\,n x + 64\,n^{2} x^{2}\right)\,P_{n}(x) = \left(27 + 216\,x + 576\,x^{2} + 512\,x^{3} + 216\,n x + 960\,n x^{2} + 1024\,n x^{3} + 192\,n^{2} x^{2} + 512\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
133235
29192576777
3278645184819214267
48134563110498304102400235345
524312960155520737280153600011796483621651
672946656699840442368013824000212336641284505653073625
7218716329629393282322432096768000222953472269746176134217728750014507

All 15 rows

Technical details
Node id
n-6ba6b6f4df801f31 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-476cf4c2bd2dc965

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:     3,      32;
  2:     9,     192,      576;
  3:    27,     864,     5184,      8192;
  4:    81,    3456,    31104,     98304,     102400;
  5:   243,   12960,   155520,    737280,    1536000,     1179648;
  6:   729,   46656,   699840,   4423680,   13824000,    21233664,    12845056;
  7:  2187,  163296,  2939328,  23224320,   96768000,   222953472,   269746176,   134217728;
  8:  6561,  559872, 11757312, 111476736,  580608000,  1783627776,  3236954112,  3221225472,  1358954496;
  9: 19683, 1889568, 45349632, 501645312, 3135283200, 12039487488, 29132587008, 43486543872, 36691771392, 13421772800;
%F Formula
The row polynomials satisfy (9+(72*n-24)*x+64*n^2*x^2)*P(n,x) = (27+(216*n+216)*x+(192*n^2+960*n+576)*x^2+(512*n^2+1024*n+512)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((27 + (216*n + 216)*x + (192*n^2 + 960*n + 576)*x^2 + (512*n^2 + 1024*n + 512)*x^3)*P[n - 1, x])/(9 + (72*n - 24)*x + 64*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; for(n = 1, N, my(i = n + 1); v[i] = ((27 + (216*n + 216)*x + (192*n^2 + 960*n + 576)*x^2 + (512*n^2 + 1024*n + 512)*x^3)*v[i - 1])/(9 + (72*n - 24)*x + 64*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

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.