Real-rooted atlas

A038281 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 25\,x + 14\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 308\,x + 196\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0121308196625
1133150826468274415625
2146417453614229612073638416390625
316105110248702608760332024021128805378249765625
4177156113528284430445407304528069725040354963847529536244140625
5194871711736129786628859161406121640178960936013666107845797742721054135046103515625
6214358881218256315297223267682474774086439371405920400872496642551006796892763883521475789056152587890625
7235794769127009219006137501478576408337724256779553837216992159429184841832242944459181223424146103116544206610467843814697265625
82593742460133011267674018906453304206416735666880142918203489602182750744204823150386680960168366448588808035671409920227271514624028925465497695367431640625
9285311670611399436338855425418676108980970531269615602470443231748804401880667479685602393576792325093085069811203241054135334401374992663475203499981325209640495651696642384185791015625
10313842837672147932360662648335526524638536142344586210288040762313323855208300689258675968123252658689431041344574458430156810695478646603520604996771929088023099876746383365345426023956485669391237529659604644775390625
1134522712143931571193964563222436184482030096820355275828071184647667867256810401483748204988329282517875741798376963204569125925207043058906892928606722162863459646489601101094124910940163821979607128883281072294696673287937147732541441490116119384765625
1237974983358324167664515802104765597700852719575628497386159299657699740851557548801625388580396466967684846910802961875648705005207703545548878511943585167572486661619455711187968423921238090711961619619495316594933766242566691643842561222320750811381761111200682555801637252902984619140625
1341772481694156517974746505248061071047741592209998039183905969037029201496112773363232024041891157654170496672888600313876343868801454073240888562694401850638670221807065601831945350320576691201398940085699349473288093041818095410176034334116804041134080100841461941938995201833481126217072640155568095557812224931322574615478515625
1445949729863572161935703589949105824893171608587782832053048980388850131840219429873426607363520670258522466727946752156393655242236520908828435210044043003801604071405074487975544320460603402366316425216041035575847180917882882848750719969584381952151070113937780989952059160324339270877184016134633910710239232027379984818174951424217795333780937113623283064365386962890625
Technical details
Node id
n-20ba9c6aa90dda9f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-64aa42accc9ceb8f

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:          121,           308,            196;
  1:         1331,          5082,           6468,           2744;
  2:        14641,         74536,         142296,         120736,           38416;
  3:       161051,       1024870,        2608760,        3320240,         2112880,          537824;
  4:      1771561,      13528284,       43044540,       73045280,        69725040,        35496384,         7529536;
  5:     19487171,     173612978,      662885916,     1406121640,      1789609360,      1366610784,       579774272,       105413504;
  6:    214358881,    2182563152,     9722326768,    24747740864,     39371405920,     40087249664,     25510067968,      9276388352,      1475789056;
  7:   2357947691,   27009219006,   137501478576,   408337724256,    779553837216,    992159429184,    841832242944,    459181223424,    146103116544,     20661046784;
  8:  25937424601,  330112676740,  1890645330420,  6416735666880,  14291820348960,  21827507442048,  23150386680960,  16836644858880,   8035671409920,   2272715146240,   289254654976;
  9: 285311670611, 3994363388554, 25418676108980, 97053126961560, 247044323174880, 440188066747968, 560239357679232, 509308506981120, 324105413533440, 137499266347520, 34999813252096, 4049565169664;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+25*x+14*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 308*x + 196*x^2.
%t Mathematica
P[0, x_] := 121 + 308*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((11 + 25*x + 14*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] = 121 + 308*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((11 + 25*x + 14*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.