Real-rooted atlas

A013611 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(5 + 26\,x + 21\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
152126
225210441676
312515756615926117576
46251050066150185220194481456976
531256562555125023152504862025408410111881376
6156253937504134375231525007293037512252303085766121308915776
7781252296875289406252025843758508543752144153025300181423518010885418031810176
8390625131250001929375001620675000850854375028588707000600362847007204354164037822859361208827064576
919531257382812512403125001215506250076576893750321622953750900544270500162097968690017020286712457942800465815429503678976
10976562541015625077519531258682187500063814078125032162295375001125680338125027016328115000425507167811253971400232905016679880978201141167095653376
114882812522558593754737304687559690039062550139632812502948210409375012382483719375037147451158125078009647432062510921350640488759173934538010553502775005422213670344486987776
12244140625123046875002842382812503979335937500376047246093752527037493750001238248371937500445769413897500011701447114809375218427012809775002752180361403165021016650032533260735582751138664195428956661682176
13122070312566650390625167958984375025865683593750271589677734375205321796367187511498020596562500482916865055625001521188124925218753549438958158843755963057449706857506830411260573309504781287882401316651544723777391194612481152873203736576
14610351562535888671875097976074218751645998046875001901127744140625159694730507812501006076802199218754829168650556250001774719479079421875496921454142238125010435350536987000625159376262746710555001673450758840460827510813066441738362270324391993252150868164509974703297150976
Technical details
Node id
n-a2a90b8c15762f85 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f594353b5992a359

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:       5,       21;
  2:      25,      210,        441;
  3:     125,     1575,       6615,        9261;
  4:     625,    10500,      66150,      185220,      194481;
  5:    3125,    65625,     551250,     2315250,     4862025,      4084101;
  6:   15625,   393750,    4134375,    23152500,    72930375,    122523030,     85766121;
  7:   78125,  2296875,   28940625,   202584375,   850854375,   2144153025,   3001814235,    1801088541;
  8:  390625, 13125000,  192937500,  1620675000,  8508543750,  28588707000,  60036284700,   72043541640,   37822859361;
  9: 1953125, 73828125, 1240312500, 12155062500, 76576893750, 321622953750, 900544270500, 1620979686900, 1702028671245, 794280046581;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (5+26*x+21*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[((5 + 26*x + 21*x^2)*P[n - 1, x])/(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] = ((5 + 26*x + 21*x^2)*v[i - 1])/(1 + 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.