Real-rooted atlas

A038303 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 4\,n x\right)\,P_{n}(x) = \left(121 + 88\,x + 16\,x^{2} + 44\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111819
212117648345
31331290415842566075
4146414259234848112641280104625
51610515856406388803097607040061441771875
61771561773044810541520681472023232004055042867229615625
7194871719920741616233940813118336059628800156119042207744131072489796875

All 15 rows

Technical details
Node id
n-a3b3b8b85a1cda73 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-83d8ce560a875699

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         11,           8;
  2:        121,         176,          48;
  3:       1331,        2904,        1584,         256;
  4:      14641,       42592,       34848,       11264,        1280;
  5:     161051,      585640,      638880,      309760,       70400,        6144;
  6:    1771561,     7730448,    10541520,     6814720,     2323200,      405504,      28672;
  7:   19487171,    99207416,   162339408,   131183360,    59628800,    15611904,    2207744,    131072;
  8:  214358881,  1247178944,  2380977984,  2308827136,  1311833600,   457949184,   97140736,  11534336,   589824;
  9: 2357947691, 15433839432, 33673831488, 38095647744, 25974305280, 11334242304, 3205644288, 570949632, 58392576, 2621440;
%F Formula
The row polynomials satisfy (11+4*n*x)*P(n,x) = (121+(44*n+88)*x+(16*n+16)*x^2)*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[((121 + (44*n + 88)*x + (16*n + 16)*x^2)*P[n - 1, x])/(11 + 4*n*x)]];
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] = ((121 + (44*n + 88)*x + (16*n + 16)*x^2)*v[i - 1])/(11 + 4*n*x)); 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.