Real-rooted atlas

A038277 → x ↦ x/(x+1) → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 - 126\,x + 378\,n x + 324\,n^{2} x^{2}\right)\,P_{n}(x) = \left(343 + 2646\,x + 6804\,x^{2} + 5832\,x^{3} + 2646\,n x + 11340\,n x^{2} + 11664\,n x^{3} + 2268\,n^{2} x^{2} + 5832\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
177279
249100829163973
3343105846123693312165475
4240198784857304261273626244006195625
51680786436010001880457228809185400068024448216484375
611764972606241050197406401203201928934000285702681616665989767205078125
7823543592950961029193452784147392031505922000699971569928166334982439182082048231279296875
857648014743607689605805552878245079044410829080001306613597184228657379507221941965946888926168066567218994140625
94035360737355910488645224996892215733299255576446408002057916415564848018049696512691171927326725623485881932819835929036800220355224609375
10282475249290545970407564571872209221573329920648391874760002881082981790728403158696889601612734497095680196822005867648013885150325760004320265344215046604766845703125
111977326743223720397208647191149066088757643300480713231062236000369738982663142412940864393209984310451390690918405051764817269632053457828754176000332660431504558089254651051409408194988250732421875
12138412872011708410305952543640565215448284046708044807488926153478000443686779195770881811721015049397765215583363607429121060870611626622720149681920511692800013971738123191439367773906883183902721955045034610237445683231353759765625
1396889010407129554448201364497390130418647538482504320768757213644407220005046937113351893762355237319564217088791030143480460083219307845131604533504340526369164101120004238093897368069939235371276318486757376177909098149531607044081301113671548928163838863372802734375
146782230728499766412249025636728686065085566716102594758502474206937151907560054955537456498398722885165716466165932811074422008726441164831536147048287404723266743168356163819520010383330048551771351041155461693070567407616871754580932704874496399967509139811794944843330077059682304004678547382354736328125
Technical details
Node id
n-eed5782960f78ff2 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c5aad9b6ec0d044d

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        7,         72;
  2:       49,       1008,        2916;
  3:      343,      10584,       61236,        93312;
  4:     2401,      98784,      857304,      2612736,       2624400;
  5:    16807,     864360,    10001880,     45722880,      91854000,       68024448;
  6:   117649,    7260624,   105019740,    640120320,    1928934000,     2857026816,     1666598976;
  7:   823543,   59295096,  1029193452,   7841473920,   31505922000,    69997156992,    81663349824,    39182082048;
  8:  5764801,  474360768,  9605805552,  87824507904,  441082908000,  1306613597184,  2286573795072,  2194196594688,   892616806656;
  9: 40353607, 3735591048, 86452249968, 922157332992, 5557644640800, 20579164155648, 48018049696512, 69117192732672, 56234858819328, 19835929036800;
%F Formula
The row polynomials satisfy (49+(378*n-126)*x+324*n^2*x^2)*P(n,x) = (343+(2646*n+2646)*x+(2268*n^2+11340*n+6804)*x^2+(5832*n^2+11664*n+5832)*x^3)*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[((343 + (2646*n + 2646)*x + (2268*n^2 + 11340*n + 6804)*x^2 + (5832*n^2 + 11664*n + 5832)*x^3)*P[n - 1, x])/(49 + (378*n - 126)*x + 324*n^2*x^2)]];
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] = ((343 + (2646*n + 2646)*x + (2268*n^2 + 11340*n + 6804)*x^2 + (5832*n^2 + 11664*n + 5832)*x^3)*v[i - 1])/(49 + (378*n - 126)*x + 324*n^2*x^2)); 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.