Real-rooted atlas

A038335 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 9\,n x\right)\,P_{n}(x) = \left(147 + 378\,x + 243\,x^{2} + 189\,n x + 243\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1215475
2441226821874896
392617144213778178732297216
4194481200037657868026613488265720517252352
540841015250987020253807034720812027900652586093442971440128
68576612113232487246379949205145827410401757741107510847773692271194342353508833280
7180108854132419593738187570506627535915733220861293142675797311366362398655683181836828256242898649939968
837822859361778070249712525197418555618006768636192361743119923504464943651627233487077387204140587147048322541865828329154986042359808
9794280046581183819096494461418033030100125672132120400481367388993310830210968587539385221096858753938521328548539606624480412641554181762559748498708200170604855296
101667988097820142891122515374037223367040128151701639636120144047858614765879050886068067665417841107585084581772309299839777246368050443327363189005160137547184727002264802453041139430128536469110784
113502775005422219907849301051394955399754029955854913484449296915801579334287274008650341136206051185868451170430907677880265370657471359777520388413620696555338518495886699835968505231693666525031096670872679866627622399146427448033280
12735582751138664122697982035135920824076073801554887421375775645803136424049749030049131272475122809034178426912624214915809812247097092270681136556532787008244700581038829863255155365448278622135400659193401982153917341681059915326390155219512302588609885731159301909010179948544
13154472377739119461516379091299342198259752437707495311506375586751304256243752150905391149031526507541908582913388183932948381716582677636786588123159917133222418088641336065172472011053372310603691845015960741050599865995803760064779422946467844205225618485326858606690498804295673577214226566773859685349896137006383104
14324391993252150868111678111757077431251614639347238336351318971003840953485921233300844366184997815268793205136901370850401400847604308028084413403151907109517271651959534155972288654671934511285416152427623497080486938491573740440901506915763647612859224875384873211211061104783048215183503324230631668031700982610631497216412848369726853881353058761991669698011332608
Technical details
Node id
n-dc548af2e1837ab1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8e2fe519da2299bf

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:           21,             54;
  2:          441,           2268,            2187;
  3:         9261,          71442,          137781,           78732;
  4:       194481,        2000376,         5786802,         6613488,          2657205;
  5:      4084101,       52509870,       202538070,       347208120,        279006525,         86093442;
  6:     85766121,     1323248724,      6379949205,     14582741040,      17577411075,      10847773692,       2711943423;
  7:   1801088541,    32419593738,    187570506627,    535915733220,     861293142675,     797311366362,     398655683181,      83682825624;
  8:  37822859361,   778070249712,   5251974185556,  18006768636192,   36174311992350,   44649436516272,   33487077387204,   14058714704832,   2541865828329;
  9: 794280046581, 18381909649446, 141803303010012, 567213212040048, 1367388993310830, 2109685875393852, 2109685875393852, 1328548539606624, 480412641554181, 76255974849870;
%F Formula
The row polynomials satisfy (7+9*n*x)*P(n,x) = (147+(189*n+378)*x+(243*n+243)*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[((147 + (189*n + 378)*x + (243*n + 243)*x^2)*P[n - 1, x])/(7 + 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] = ((147 + (189*n + 378)*x + (243*n + 243)*x^2)*v[i - 1])/(7 + 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

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.