Real-rooted atlas

A038326 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(23 + 24\,n x\right)\,P_{n}(x) = \left(529 + 1104\,x + 576\,x^{2} + 552\,n x + 576\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1234871
2529220817284465
3121677617611923255296262871
4279841233606454846725087232165888014846689
564363436716184021024576029251584019077120047775744814906727
61480358891853666784725347872013455728640131632128006593052672133772083243804896337
7340482544749740058704233562014784541593077760706425753600530740740096215373053952366917713922317531295735
87831098528113074529716487162568453376199306252615683249558466560032552098725888198143209635846751285936128990677827584121082925790657
9180115266146333830345641392211807381406976687606571524096134531720515584016845711090647041367188146487296698758094389248205070310309888264180754022406262368392043143
104142651121364986455327750224060894622154505602259278735007744051570492864307200774902710169763847861331842301952053571453903175680235830856856371206076157342515200697437190619136321190445780494129
1195280975791392721873197920806672171181548945443520714496899946199040186390495638138880032675064278825041923977833912204787712338839445937586176019888402261553971207686339038281728001764516092266414081826017371802828816358330125343145239
12219146244320203215488184205584219524724610750894241152219112382650167705606430472099515791360012883311058508159385618298035996142023475218703937415754756915213722997560472240128070714319152191897600243503220732765143045039807946175807488474764516668735488828173337053144121505
135040363619364673831367472564558068030412842351041067073676865514602412400143974421363457308391351296004815137508117424570368781587537549495002726493207954788511205294088206352541162399596544528589535662634434560024269154333032259256327534512879532832194561419545904839519109121227083673851500953641712742436112539588519
141159283632453874980933871243522130608137634460308626863314366081917791088799349669068868790332533020151173120172274919734867856850944314588983863671738597376428756592027151544352768440407586375715445014528340411660966736575856640195366692380909687013376808704382403190655549442285468906791625765683239512094298018330705923155358018475288166402091562887444183356916241
Technical details
Node id
n-ad26ad5916dd6608 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-979f2fd21312bce1

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:            23,             48;
  2:           529,           2208,            1728;
  3:         12167,          76176,          119232,           55296;
  4:        279841,        2336064,         5484672,         5087232,          1658880;
  5:       6436343,       67161840,       210245760,       292515840,        190771200,         47775744;
  6:     148035889,     1853666784,      7253478720,     13455728640,      13163212800,       6593052672,       1337720832;
  7:    3404825447,    49740058704,    233562014784,    541593077760,     706425753600,     530740740096,     215373053952,     36691771392;
  8:   78310985281,  1307452971648,   7162568453376,  19930625261568,   32495584665600,   32552098725888,   19814320963584,   6751285936128,    990677827584;
  9: 1801152661463, 33830345641392, 211807381406976, 687606571524096, 1345317205155840, 1684571109064704, 1367188146487296, 698758094389248, 205070310309888, 26418075402240;
%F Formula
The row polynomials satisfy (23+24*n*x)*P(n,x) = (529+(552*n+1104)*x+(576*n+576)*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[((529 + (552*n + 1104)*x + (576*n + 576)*x^2)*P[n - 1, x])/(23 + 24*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] = ((529 + (552*n + 1104)*x + (576*n + 576)*x^2)*v[i - 1])/(23 + 24*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.