Real-rooted atlas

A038279 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(81 - 18\,x + 54\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(729 + 486\,x + 108\,x^{2} + 8\,x^{3} + 486\,n x + 180\,n x^{2} + 16\,n x^{3} + 36\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19817
28114436261
372919449721283773
465612332817496460840052393
559049262440262440103680180001152706761
65314412834352354294018662404860006220831369326317
747829692976069644641044293932801020600019595521975688192120949301

All 15 rows

Technical details
Node id
n-904a6a04e6b94575 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-52a22fb5e4297724

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:         9,          8;
  2:        81,        144,         36;
  3:       729,       1944,        972,        128;
  4:      6561,      23328,      17496,       4608,        400;
  5:     59049,     262440,     262440,     103680,      18000,      1152;
  6:    531441,    2834352,    3542940,    1866240,     486000,     62208,      3136;
  7:   4782969,   29760696,   44641044,   29393280,   10206000,   1959552,    197568,     8192;
  8:  43046721,  306110016,  535692528,  423263232,  183708000,  47029248,   7112448,   589824,   20736;
  9: 387420489, 3099363912, 6198727824, 5714053632, 2976069600, 952342272, 192036096, 23887872, 1679616, 51200;
%F Formula
The row polynomials satisfy (81+(54*n-18)*x+4*n^2*x^2)*P(n,x) = (729+(486*n+486)*x+(36*n^2+180*n+108)*x^2+(8*n^2+16*n+8)*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[((729 + (486*n + 486)*x + (36*n^2 + 180*n + 108)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(81 + (54*n - 18)*x + 4*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] = ((729 + (486*n + 486)*x + (36*n^2 + 180*n + 108)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(81 + (54*n - 18)*x + 4*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

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.