Real-rooted atlas

A038323 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 9\,x + 9\,n\right)\,P_{n}(x) = \left(400 + 360\,x + 81\,x^{2} + 180\,n + 81\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
129938
2760522811363
31880020520704772947096
44480006768003693608456465611585285
510400000201600001522800055404009513455904952338794
623680000056160000054432000027410400074795400102745265314411702425367
753120000001491840000017690400000114307200004317138000942422040107882523478296954723745532

All 15 rows

Technical details
Node id
n-5984326d454ddba1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0a6d3ccfdf16abbe

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:            29,             9;
  2:           760,           522,             81;
  3:         18800,         20520,           7047,            729;
  4:        448000,        676800,         369360,          84564,          6561;
  5:      10400000,      20160000,       15228000,        5540400,        951345,         59049;
  6:     236800000,     561600000,      544320000,      274104000,      74795400,      10274526,       531441;
  7:    5312000000,   14918400000,    17690400000,    11430720000,    4317138000,     942422040,    107882523,      4782969;
  8:  117760000000,  382464000000,   537062400000,   424569600000,  205752960000,   62166787200,  11309064480,   1109648808,    43046721;
  9: 2585600000000, 9538560000000, 15489792000000, 14500684800000, 8597534400000, 3333197952000, 839251627200, 130862031840, 11235194181, 387420489;
%F Formula
The row polynomials satisfy (9*n+11+9*x)*P(n,x) = (180*n+400+(81*n+360)*x+81*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[((180*n + 400 + (81*n + 360)*x + 81*x^2)*P[n - 1, x])/(9*n + 11 + 9*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] = ((180*n + 400 + (81*n + 360)*x + 81*x^2)*v[i - 1])/(9*n + 11 + 9*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.