Real-rooted atlas

A038320 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 28\,x + 17\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 374\,x + 289\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0121374289784
1133161719537491321952
2146419050820981421617283521614656
31610511244485384659059447304593655141985717210368
4177156116427202634687351307840601515906159371056224137569481890304
519487171210815759977418519251759315538908257853607856637185859281341033867313492928512
6214358881265025525614335471612443096395288559816727010583046135281778083772361098032246975757441377801998336
7235794769132796908793202744527084731109052212169484371194626193039184622698676764476178743525958869059998665911858787649710578455953408
8259374246014008511074702787737247405114888565347603107213471901057624686206164742136110230906553929285156037982999266245130446664146702015993900449296196766695424
92853116706114850298400387374795785484451737689550882455371040430000301162097838490974179596938675877819825636087596901531980970405215789202318087535243935261954329342718963076338293509467471872
10313842837672158203580804644494730436839474254861134129426088622167095004952191384495440122439511326508693116523396792712558165055537202337209534724901995851540160997272889857144523890312607556582622237229761232218265089212416
11345227121439316935926712553416431495678913162364451421805079181408107766065078653917099785599218798071599558204450841247429022631596948144588363986844191712414152463516925557674203341083190343234581573514402548331497992385582399045780329059376502111422497947648
123797498335832418216405490255578825375278793855795102319905271108522168485959740221121670259296646977437415537829149543567867274434384978951328563711101342329000920338235589587632130694295456828631702828091660418541070726637264152534541368983711525305017067514298168377826559400929182059119829942534144
13417724816941565196836207563726455104759170000758619570156898697477742153252728939610331681511059278394675127717128486020107496541089556602091901908711515587475960212040735978510514787136598835940859750075344846193326976849226481916747378453528389399775294104051258376639080699295852778234138230115328528624230515098157935097655355238390956032
144594972986357216111362115020810570721316972422866679788094981647467354481680477066911142848646620176948454314802043473650135395389193912317521106885352748437025192019244711246648961915270264371790863056446512802860022101154883375911224109277216346950772144155249133590112940578207382476282606769202320244484604164250148908050378645706572757956848661191875666868481142734349946674946768896
Technical details
Node id
n-382f3b939374e875 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-94bd4dafe00b9a6d

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,           374,            289;
  1:         1331,          6171,           9537,            4913;
  2:        14641,         90508,         209814,          216172,           83521;
  3:       161051,       1244485,        3846590,         5944730,         4593655,          1419857;
  4:      1771561,      16427202,       63468735,       130784060,       151590615,         93710562,         24137569;
  5:     19487171,     210815759,      977418519,      2517593155,      3890825785,       3607856637,       1858592813,        410338673;
  6:    214358881,    2650255256,    14335471612,     44309639528,     85598167270,     105830461352,      81778083772,      36109803224,       6975757441;
  7:   2357947691,   32796908793,   202744527084,    731109052212,   1694843711946,    2619303918462,    2698676764476,    1787435259588,     690599986659,    118587876497;
  8:  25937424601,  400851107470,  2787737247405,  11488856534760,  31072134719010,   57624686206164,   74213611023090,   65539292851560,   37982999266245,  13044666414670,   2015993900449;
  9: 285311670611, 4850298400387, 37479578548445, 173768955088245, 537104043000030, 1162097838490974, 1795969386758778, 1982563608759690, 1531980970405215, 789202318087535, 243935261954329, 34271896307633;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+28*x+17*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 374*x + 289*x^2.
%t Mathematica
P[0, x_] := 121 + 374*x + 289*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((11 + 28*x + 17*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 + 374*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((11 + 28*x + 17*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.