Real-rooted atlas

A147716 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
118927
232432481729
358328748437472919683
4104976209952157464524886561531441
518895684723920472392023619605904905904914348907
63401222410203667212754584085030560318864606377292531441387420489
7612220032214277011232141551682678462640133923132040176939666961566478296910460353203
8110199605764407984230477139724032771397240324821232752019284931008482123275268874753643046721282429536481
91983592903688926168066561785233613312208277254886415620794116487810397058242603465686085578855041669735688023874204897625597484987
1035704672266241785233613312040167756299520535570083993604686238234944028117429409664117155955873603347313024960627621192180697356880203486784401205891132094649
116426841007923235347625543577688369063858944013255359578841601325535957884160927875170518912463937585259456165691994735520414229986838806903833113980690383311398313810596095559060566555523
1211568313814261766940988288557056190877177935319043181286298921984035789470862872320286315766902978561670175306934041671578941725744642236841928929520497075984206560745613976309846778308875544282429536481150094635296999121
13208229648656711681353492716268625924060478148805877767444209939477442569305262424346803208374736181912122885583157454608081922791578727304040961046842022739015362907894507608376058157890152167527930621384386486608851153655425418658283294052555153018976267
143748133675820810242623693573074567168852700411249234329617054008224984686592234492613093539440642344926130935394406417586945982015458048100496834182945474564396736495503864512146557883183462150436639470795865537666617219628846432832715245360580464055018873890822876792454961109418989131512359209
Technical details
Node id
n-acb5f60cda3bf411 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-11aec07ce60b6d61

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:           18,            9;
  2:          324,          324,            81;
  3:         5832,         8748,          4374,           729;
  4:       104976,       209952,        157464,         52488,          6561;
  5:      1889568,      4723920,       4723920,       2361960,        590490,        59049;
  6:     34012224,    102036672,     127545840,      85030560,      31886460,      6377292,       531441;
  7:    612220032,   2142770112,    3214155168,    2678462640,    1339231320,    401769396,     66961566,     4782969;
  8:  11019960576,  44079842304,   77139724032,   77139724032,   48212327520,  19284931008,   4821232752,   688747536,   43046721;
  9: 198359290368, 892616806656, 1785233613312, 2082772548864, 1562079411648, 781039705824, 260346568608, 55788550416, 6973568802, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (18+9*x)*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[(18 + 9*x)*P[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] = (18 + 9*x)*v[i - 1]); 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.