Real-rooted atlas

A038277 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 11\,n x\right)\,P_{n}(x) = \left(49 + 154\,x + 121\,x^{2} + 77\,n x + 121\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
172229
249308363720
334332347623532416524
424013018410672214907273205361584
5168072641101245090260876025621759663067663248
611764922185241307344536522640538056754058485212400927158723712
7823543181179461281197614474023408788260259943288746076454231558973683231161280
85764801144943568119578443650109062081230356435018560805648170140718448730252608192922992964895323392
94035360711414305981076205992452614515184155024910810292332688956357295508724275002957152121541485527235794769101289335387392
1028247524988777935409416802433552614515184018086239594504092657645384625267140267064167356668804253951993445165056338370028531167061125389989167104
11197732674368359010258805659763755506414708646019894863553950525224397824289629113960111812352216158744010918476783175563546690272450219689986370473423740047332496294944500736
12138412872015220142601526767542015542472653728069602088960673164756302692773891361348075954415652207517231466899222928801244668551779307327628600922697942755974287594163975888407995688973739640261511884800
1396889010407395860813948655986029401302430114892543336211217134731102571693130303014221752498740740347631473446772479712417304182652967614047924170355065027988504263597878130855344609029043712760769660943483317970015034186249852409614336
1467822307284929841815205356457219240110633383193267902244820699279203648045780658529966154842146810957406925814406282548147159686815968316665137637933931373895927406857183544581480114274607923894948641819252777133862074736516106147333256962475037486153581549956895440896
Technical details
Node id
n-ff6375f366f97f27 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-38646c70532f51c4

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:        7,         22;
  2:       49,        308,         363;
  3:      343,       3234,        7623,        5324;
  4:     2401,      30184,      106722,      149072,        73205;
  5:    16807,     264110,     1245090,     2608760,      2562175,       966306;
  6:   117649,    2218524,    13073445,    36522640,     53805675,     40584852,     12400927;
  7:   823543,   18117946,   128119761,   447402340,    878826025,    994328874,    607645423,    155897368;
  8:  5764801,  144943568,  1195784436,  5010906208,  12303564350,  18560805648,  17014071844,   8730252608,   1929229929;
  9: 40353607, 1141430598, 10762059924, 52614515184, 155024910810, 292332688956, 357295508724, 275002957152, 121541485527, 23579476910;
%F Formula
The row polynomials satisfy (7+11*n*x)*P(n,x) = (49+(77*n+154)*x+(121*n+121)*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[((49 + (77*n + 154)*x + (121*n + 121)*x^2)*P[n - 1, x])/(7 + 11*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] = ((49 + (77*n + 154)*x + (121*n + 121)*x^2)*v[i - 1])/(7 + 11*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.