Real-rooted atlas

A027466 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x + n\right)\,P_{n}(x) = \left(16 + 16\,x + 4\,x^{2} + 8\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
112416
21289616240
312801536576643456
4122882048012288307225648384
511468824576020480081920153601024663552
610485762752512294912016384004915207372840968957952
7943718429360128385351682752512011468800275251234406416384119439360

All 15 rows

Technical details
Node id
n-eb1c168b96e7421e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-49d880ced4b36b3a

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:        12,          4;
  2:       128,         96,         16;
  3:      1280,       1536,        576,         64;
  4:     12288,      20480,      12288,       3072,        256;
  5:    114688,     245760,     204800,      81920,      15360,       1024;
  6:   1048576,    2752512,    2949120,    1638400,     491520,      73728,      4096;
  7:   9437184,   29360128,   38535168,   27525120,   11468800,    2752512,    344064,    16384;
  8:  83886080,  301989888,  469762048,  411041792,  220200960,   73400320,  14680064,  1572864,   65536;
  9: 738197504, 3019898880, 5435817984, 5637144576, 3699376128, 1585446912, 440401920, 75497472, 7077888, 262144;
%F Formula
The row polynomials satisfy (n+1+x)*P(n,x) = (8*n+16+(4*n+16)*x+4*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[((8*n + 16 + (4*n + 16)*x + 4*x^2)*P[n - 1, x])/(n + 1 + 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] = ((8*n + 16 + (4*n + 16)*x + 4*x^2)*v[i - 1])/(n + 1 + 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

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.