Real-rooted atlas

A130123 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-3 + 4\,x + 4\,n\right)\,P_{n}(x) = \left(2 + 16\,x + 32\,x^{2} + 8\,n + 32\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
110818
23616064260
310486419205123400
427233281382420480409642000
5672108806656018432020480032768500000
61600322562611201064960221184019660802621445800000
73712896009031684874240149094402477260818350080209715266000000

All 15 rows

Technical details
Node id
n-8a1a31d1b19d37ec · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-236e6d52004bfa9f

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:    10,      8;
  2:    36,    160,      64;
  3:   104,    864,    1920,      512;
  4:   272,   3328,   13824,    20480,      4096;
  5:   672,  10880,   66560,   184320,    204800,      32768;
  6:  1600,  32256,  261120,  1064960,   2211840,    1966080,     262144;
  7:  3712,  89600,  903168,  4874240,  14909440,   24772608,   18350080,    2097152;
  8:  8448, 237568, 2867200, 19267584,  77987840,  190840832,  264241152,  167772160,   16777216;
  9: 18944, 608256, 8552448, 68812800, 346816512, 1123024896, 2290089984, 2717908992, 1509949440, 134217728;
%F Formula
The row polynomials satisfy (4*n-3+4*x)*P(n,x) = (8*n+2+(32*n+16)*x+32*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 + 2 + (32*n + 16)*x + 32*x^2)*P[n - 1, x])/(4*n - 3 + 4*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 + 2 + (32*n + 16)*x + 32*x^2)*v[i - 1])/(4*n - 3 + 4*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.