Real-rooted atlas

A038323 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(38 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1381856
2144413683243136
35487277976369365832175616
42085136395078428071368864641049769834496
57923516818766224017778528084214080199454401889568550731776
63010936384855739814410133760960640027008022737801604308215043401222430840979456
71144155825923793779843845391160830724256179603202016085075205729926003290472515846122200321727094849536
8434779213849616475843893248273152148756482587757198745615322246571520580632501657613751822407681861148897281101996057696717311574016
91652161012628487043423264363521334543355353088147502160328499210480416654919684964407889172481567707754475523182564614348837688265169921983592903685416169448144896
1062782118479882242973889822731264063390809379271680800726013211852806637597214782464037729499957710848148932236675174404031248511508480716077038228480753765303398403570467226624303305489096114176
11238572050223552512124308594590166835229441509245039513604183793419031930880396359376539867136026284884970538557441245073498604457984421265469452636160997734006598348801575369484102656014924553007288326426841007923216985107389382393856
129065737908494995456515315628482873425921342532821573801820162119788665642844979202259248446277242675201712272506652226027529462558589393880678438419410814080417792113741676752211763202394561615836037120340279808566173696293063949961297921156831381426176951166013805414055936
13344498040522809827328212138267058782893670460291928532496190832641047175600827565419724812400763694010643128320105732827285774957199366677862775943681507328316319815702595439820811237677663118522204162957283595557505843205603274181056326860872386795640440586245714747024245309442082296486567116853265296773103187132416
141309092553986677343846486813506211748076486656267294216494066446024704506452199672968002994176659720628521366214426624624998490178136413667328444077874600254820237312240403059933972534263808996407419463175635435523146549745673186217164874523546608049147248641283659176023813062656152012270844925231104110778173085370613763748133675820810242982856619293778479415296
Technical details
Node id
n-7a7add241daaaebf · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-5d2d61b0db2d0eeb

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:              38,              18;
  2:            1444,            1368,              324;
  3:           54872,           77976,            36936,             5832;
  4:         2085136,         3950784,          2807136,           886464,           104976;
  5:        79235168,       187662240,        177785280,         84214080,         19945440,         1889568;
  6:      3010936384,      8557398144,      10133760960,       6400270080,       2273780160,       430821504,        34012224;
  7:    114415582592,    379377984384,     539116083072,     425617960320,     201608507520,     57299260032,      9047251584,      612220032;
  8:   4347792138496,  16475843893248,   27315214875648,   25877571987456,   15322246571520,   5806325016576,   1375182240768,   186114889728,   11019960576;
  9: 165216101262848, 704342326436352, 1334543355353088, 1475021603284992, 1048041665491968, 496440788917248, 156770775447552, 31825646143488, 3768826516992, 198359290368;
%F Formula
The row polynomials satisfy P(n,x) = (38+18*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[(38 + 18*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] = (38 + 18*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.