Real-rooted atlas

A013622 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(8 + 36\,x + 28\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
182836
2644487841296
35125376188162195246656
44096573443010567024646146561679616
532768573440401408014049280245862401721036860466176
62621445505024481689602247884805900697608260976644818903042176782336
720971525138022453949235231470387201101463552023130734592269858570241349292851278364164096
8167772164697620485754585088402820956161762341683204934556712968635474247688635474247683778019983362821109907456
91342177284227858432591900180484833851473922537772023808888220208332820725138194432310877072916482720174388019210578455953408101559956668416
1010737418243758096384059190018048055244016844803383696031744014211523333324841450276388864082900552777728010880697552076808462764762726402961967666954243656158440062976
11858993459233071248179257874684313606076841852928042537892970496020843567555543047295248644440064182381216111001603191671281942528037236164955996160260653154691973128293509467471872131621703842267136
1268719476736288621802291255559696941056648196464312320510454715645952028585464076173312116723978311041024350171934933123072766001107666206720119155727859187712012511351425214709767961769088772997122322182650892124164738381338321616896
13549755813888250138895319045252916801699846741243228848128589858782524211203716110329902530561734184820621180928606964687217413324815932823039457099776309804892433888051204337268494074432716841401199261619585024241506995692780912646502111422497947648170581728179578208256
144398046511104215504279044096490272234825318468638112875544576660641836427116544462449285498981580824278587488696532992971143499547861319682974126967365325291526939629590519092346881214435178340841160704154564477243379784089613524391758795731107847282364793197701365761820591198299425341446140942214464815497216
Technical details
Node id
n-8c340f191f09d9f8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8d10fbbacee008e8

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:         8,         28;
  2:        64,        448,         784;
  3:       512,       5376,       18816,        21952;
  4:      4096,      57344,      301056,       702464,        614656;
  5:     32768,     573440,     4014080,     14049280,      24586240,      17210368;
  6:    262144,    5505024,    48168960,    224788480,     590069760,     826097664,      481890304;
  7:   2097152,   51380224,   539492352,   3147038720,   11014635520,   23130734592,    26985857024,    13492928512;
  8:  16777216,  469762048,  5754585088,  40282095616,  176234168320,  493455671296,   863547424768,   863547424768,   377801998336;
  9: 134217728, 4227858432, 59190018048, 483385147392, 2537772023808, 8882202083328, 20725138194432, 31087707291648, 27201743880192, 10578455953408;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (8+36*x+28*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[((8 + 36*x + 28*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] = ((8 + 36*x + 28*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.