Real-rooted atlas

A038323 → 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

\(\left(20 + 9\,n x\right)\,P_{n}(x) = \left(400 + 360\,x + 81\,x^{2} + 180\,n x + 81\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1201838
24007202431363
380002160014580291647096
4160000576000583200233280328051585285
53200000144000001944000011664000328050035429452338794
6640000003456000005832000004665600001968300004251528037200871702425367
7128000000080640000001632960000016329600000918540000029760696005208121803826375254723745532
8256000000001843200000004354560000005225472000003674160000001587237120004166497440061222003203874204891742237507209
951200000000041472000000001119744000000015676416000000132269760000007142567040000249989846400055099802880069735688020387420489055027105425710
101024000000000092160000000000279936000000000447897600000000440899200000000285702681600000124994923200000367332019200006973568802000774840978000383546284111726350371128411
11204800000000000202752000000000068428800000000001231718400000000013856832000000000104757649920000005499776620800000202032610560000051139504548000085232507580000843801825042037657271530853850525862425728
124096000000000000442368000000000001642291200000000003284582400000000004157049600000000003591690854400000002199910648320000009697565306880000030683702728800000681860060640000010125621900504009037745167392036715839742531671469837901698573
138192000000000000095846400000000000038817792000000000008539914240000000000120092544000000000001167299527680000000081710966937600000004202278299648000000159555254189760000044320903941600000087755389804368000117490687176096009546118333057803558612159660651656958347998479986
141638400000000000000206438400000000000009057484800000000000021737963520000000000033625912320000000000036315985305600000000028598838428160000000016809113198592000000074459118621888000000248197062072960000006142877286305760000109657974697689600013364565666280920099641140470496803431518868244151590397450508583339295
Technical details
Node id
n-a90658b17f2703fe · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-65ecce61f1255cba

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:           20,            18;
  2:          400,           720,            243;
  3:         8000,         21600,          14580,           2916;
  4:       160000,        576000,         583200,         233280,          32805;
  5:      3200000,      14400000,       19440000,       11664000,        3280500,        354294;
  6:     64000000,     345600000,      583200000,      466560000,      196830000,      42515280,       3720087;
  7:   1280000000,    8064000000,    16329600000,    16329600000,     9185400000,    2976069600,     520812180,     38263752;
  8:  25600000000,  184320000000,   435456000000,   522547200000,   367416000000,  158723712000,   41664974400,   6122200320,   387420489;
  9: 512000000000, 4147200000000, 11197440000000, 15676416000000, 13226976000000, 7142567040000, 2499898464000, 550998028800, 69735688020, 3874204890;
%F Formula
The row polynomials satisfy (20+9*n*x)*P(n,x) = (400+(180*n+360)*x+(81*n+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[((400 + (180*n + 360)*x + (81*n + 81)*x^2)*P[n - 1, x])/(20 + 9*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] = ((400 + (180*n + 360)*x + (81*n + 81)*x^2)*v[i - 1])/(20 + 9*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.