Real-rooted atlas

A038221 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(15 + 27\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1152742
22258107291764
3337518225328051968374088
45062536450098415011809805314413111696
5759375683437524603750442867503985807514348907130691232
6113906251230187505535843751328602500179361337512914016303874204895489031744
71708593752152828125116252718753487581562562776468125677985855754067915134510460353203230539333248
825628906253690562500023250543750083701957500018832940437502711943423000244074908070012552423843602824295364819682651996416
938443359375622782421875448403343750018832940437500508489391812509152809052625010983370863150084728860944300381279874249357625597484987406671383849472
10576650390625103797070312508407562695312540356300937500012712234795312502745842715787500411876407368125042364430472150002859599056870125114383962274805020589113209464917080198121677824
11864975585937517126516601562515413864941406258323487068359375299645534460937507551067468415625013591921443148125017475327569761875015727794812785687594366768876714125339720367956170855559060566555523717368321110468608
12129746337890625280252089843750027744956894531250166469741367187500674202452537109375194170306330687500040775764329444375006291117925114275000707750766575355937556620061326028475003057483311605537650100063090197999414015009463529699912130129469486639681536
1319461950683593754554096459960937549184241767578125032461599566601562501460771980497070312547329012168105078125113589629203452187500204461332566213937500276022798964388815625276022798964388815625198736415254359947250975615129430494286502926845388291482859540525551530189762671265437718438866624512
1429192926025390625735661735839843750860724230932617187561972144627148437500306762115904384765625110434361725578515625029817277665906199218756133839976986418125000966079796375360854687511592957556504330256250104336618008538972306256829305906013460005500307318765770605700247585103658213398501607010941898913151235920953148384174432398229504
Technical details
Node id
n-ab0ac67f032e783e · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dca555f310728b4d

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:          15,           27;
  2:         225,          810,           729;
  3:        3375,        18225,         32805,          19683;
  4:       50625,       364500,        984150,        1180980,         531441;
  5:      759375,      6834375,      24603750,       44286750,       39858075,       14348907;
  6:    11390625,    123018750,     553584375,     1328602500,     1793613375,     1291401630,       387420489;
  7:   170859375,   2152828125,   11625271875,    34875815625,    62776468125,    67798585575,     40679151345,    10460353203;
  8:  2562890625,  36905625000,  232505437500,   837019575000,  1883294043750,  2711943423000,   2440749080700,  1255242384360,   282429536481;
  9: 38443359375, 622782421875, 4484033437500, 18832940437500, 50848939181250, 91528090526250, 109833708631500, 84728860944300, 38127987424935, 7625597484987;
%F Formula
The row polynomials satisfy P(n,x) = (15+27*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[(15 + 27*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] = (15 + 27*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.