Real-rooted atlas

A013610 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-5 + 9\,x + 9\,n\right)\,P_{n}(x) = \left(16 + 72\,x + 81\,x^{2} + 36\,n + 81\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113922
28823481403
3496237631597296760
425601785642768379086561107653
512544115200401760641520426465590491656538
659392677376311040072316808660520460582253144124876631
727443237416962133734465318400113898960109122552483611314782969366837484
8124518419759104134701056512096256117573120016401450241309470624497428776430467215333623945
95570560100859904800243712363692851210369949184190468454402214195782415152445792503646635738742048976678687774
1024641536501350400453869568024007311360818308915201866590853122857026816002846823148801704650151605036466357034867844011092263435419
11108003328243951206424816844800149776957440594180956160162025165209630798749076484040652211200352294364664018751151667604986101693433138105960915440151087088
124697620481166435942413173365145689340641280040439778508801283430865305629164529737728475180700037125454880485120042275323759680202512438010084895445299004282429536481216851407678477
13203004313654962159616682365026304513761240678426132137574400946290817105922502690187345924874642827591686949517738042887091344630656004946212879882562153995931561764773059166528925418658283293028751065922530
14872415232025578543513634626160558082865933110476816183479081369665852986687488019872107159224324504842337222656767756245345689697293248332600328935094234626560566566202604729622616957281398484626195807558782287679245496142099639816323167
Technical details
Node id
n-e1a9c0039721e0c6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0396f0c135a59aec

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:      13,         9;
  2:      88,       234,        81;
  3:     496,      2376,      3159,        729;
  4:    2560,     17856,     42768,      37908,        6561;
  5:   12544,    115200,    401760,     641520,      426465,       59049;
  6:   59392,    677376,   3110400,    7231680,     8660520,     4605822,      531441;
  7:  274432,   3741696,  21337344,   65318400,   113898960,   109122552,    48361131,     4782969;
  8: 1245184,  19759104, 134701056,  512096256,  1175731200,  1640145024,  1309470624,   497428776,   43046721;
  9: 5570560, 100859904, 800243712, 3636928512, 10369949184, 19046845440, 22141957824, 15152445792, 5036466357, 387420489;
%F Formula
The row polynomials satisfy (9*n-5+9*x)*P(n,x) = (36*n+16+(81*n+72)*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[((36*n + 16 + (81*n + 72)*x + 81*x^2)*P[n - 1, x])/(9*n - 5 + 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] = ((36*n + 16 + (81*n + 72)*x + 81*x^2)*v[i - 1])/(9*n - 5 + 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.