Real-rooted atlas

A038329 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(15 + 42\,x + 27\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 225 + 810\,x + 729\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02258107291764
1337518225328051968374088
25062536450098415011809805314413111696
3759375683437524603750442867503985807514348907130691232
4113906251230187505535843751328602500179361337512914016303874204895489031744
51708593752152828125116252718753487581562562776468125677985855754067915134510460353203230539333248
625628906253690562500023250543750083701957500018832940437502711943423000244074908070012552423843602824295364819682651996416
738443359375622782421875448403343750018832940437500508489391812509152809052625010983370863150084728860944300381279874249357625597484987406671383849472
8576650390625103797070312508407562695312540356300937500012712234795312502745842715787500411876407368125042364430472150002859599056870125114383962274805020589113209464917080198121677824
9864975585937517126516601562515413864941406258323487068359375299645534460937507551067468415625013591921443148125017475327569761875015727794812785687594366768876714125339720367956170855559060566555523717368321110468608
10129746337890625280252089843750027744956894531250166469741367187500674202452537109375194170306330687500040775764329444375006291117925114275000707750766575355937556620061326028475003057483311605537650100063090197999414015009463529699912130129469486639681536
1119461950683593754554096459960937549184241767578125032461599566601562501460771980497070312547329012168105078125113589629203452187500204461332566213937500276022798964388815625276022798964388815625198736415254359947250975615129430494286502926845388291482859540525551530189762671265437718438866624512
1229192926025390625735661735839843750860724230932617187561972144627148437500306762115904384765625110434361725578515625029817277665906199218756133839976986418125000966079796375360854687511592957556504330256250104336618008538972306256829305906013460005500307318765770605700247585103658213398501607010941898913151235920953148384174432398229504
1343789389038085937511823135040283203125148971501507568359375116197771175903320312562746796434987792968752484773138825516601562574543194164765498046875172514249352743009765625310525648834937417578125434735908368912384609375469514781038425375378125384148457213257125309375230489074327954275185625957416154900733143078752461927255459028082202529543127065508336986432232232135326160725639168
14656840835571289062518917016064453125000025537971687011718750002145189621708984375000012549359286997558593750054213232119829453125000017890366599543719531250004600379982739813593750000931576946504812252734375014905231144076996043750000187805912415370150151250001843912594623634201485000013829344459677256511137500765932923920586514463000029543127065508336986430007090350495722000876743207976644307687250986336193753749683698750476845056
Technical details
Node id
n-950319b6c870b4e9 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-dca555f310728b4d

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           225,             810,              729;
  1:          3375,           18225,            32805,            19683;
  2:         50625,          364500,           984150,          1180980,            531441;
  3:        759375,         6834375,         24603750,         44286750,          39858075,          14348907;
  4:      11390625,       123018750,        553584375,       1328602500,        1793613375,        1291401630,          387420489;
  5:     170859375,      2152828125,      11625271875,      34875815625,       62776468125,       67798585575,        40679151345,        10460353203;
  6:    2562890625,     36905625000,     232505437500,     837019575000,     1883294043750,     2711943423000,      2440749080700,      1255242384360,       282429536481;
  7:   38443359375,    622782421875,    4484033437500,   18832940437500,    50848939181250,    91528090526250,    109833708631500,     84728860944300,     38127987424935,     7625597484987;
  8:  576650390625,  10379707031250,   84075626953125,  403563009375000,  1271223479531250,  2745842715787500,   4118764073681250,   4236443047215000,   2859599056870125,  1143839622748050,   205891132094649;
  9: 8649755859375, 171265166015625, 1541386494140625, 8323487068359375, 29964553446093750, 75510674684156250, 135919214431481250, 174753275697618750, 157277948127856875, 94366768876714125, 33972036795617085, 5559060566555523;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (15+42*x+27*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 810*x + 729*x^2.
%t Mathematica
P[0, x_] := 225 + 810*x + 729*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15 + 42*x + 27*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] = 225 + 810*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15 + 42*x + 27*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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.