Real-rooted atlas

A038221 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + n x\right)\,P_{n}(x) = \left(12 + 12\,x + 3\,x^{2} + 6\,n x + 3\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16612
2367227135
32166484861081458
4129651845832259240515309
57776388805832038880121501458157464
6466562799365248804665602187005248851031594323
7279936195955244089924898880306180011022482143261749615943230
816796161343692835271936470292483674160017635968514382483980859049157837977
91007769690699264272097792423263232396809280238085568925888322267481631886461968301549681956
106046617660466176020407334403627970560396809280028570268161388832480453496320956593801180980064953915109399071
113627970563990767616149653785602993075712037413446400314272949761833258873674826892802104506360389723400428695742125764146444944842
12217678233626121388032107750725632239446056960336721017600323252176896219991064832107750725632378811144809353361600154330466415305500869087331412147682405
1313060694016169789022208764050599936186767924428829182488192003151708724736245132900812814007594332165909453858881823905512004012592126459691453125388811742232052213556617751088
14783641640961097098297344534835419955214262277865472245132900812802941594809753625738954585344168091131985928273235402432306416126016084264434654416713606873622633009308187492384871744535129635157244779
Technical details
Node id
n-141ca264cc04332e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e620d4a356abb37c

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:        6,        6;
  2:       36,       72,        27;
  3:      216,      648,       486,       108;
  4:     1296,     5184,      5832,      2592,       405;
  5:     7776,    38880,     58320,     38880,     12150,      1458;
  6:    46656,   279936,    524880,    466560,    218700,     52488,     5103;
  7:   279936,  1959552,   4408992,   4898880,   3061800,   1102248,   214326,    17496;
  8:  1679616, 13436928,  35271936,  47029248,  36741600,  17635968,  5143824,   839808,   59049;
  9: 10077696, 90699264, 272097792, 423263232, 396809280, 238085568, 92588832, 22674816, 3188646, 196830;
%F Formula
The row polynomials satisfy (2+n*x)*P(n,x) = (12+(6*n+12)*x+(3*n+3)*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[((12 + (6*n + 12)*x + (3*n + 3)*x^2)*P[n - 1, x])/(2 + 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] = ((12 + (6*n + 12)*x + (3*n + 3)*x^2)*v[i - 1])/(2 + 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

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.