Real-rooted atlas

A038321 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(32 + 42\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1024 + 2688\,x + 1764\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01024268817645476
13276812902416934474088405224
210485765505024108380169483264311169629986576
3335544322202009605780275207586611204978713601306912322219006624
410737418248455716864277453209604855431168047795650560250927165445489031744164206490176
5343597383683156800962561242990379008271904145408035687419084802810384252928122954311065623053933324812151280273024
610995116277761154487209164853034256171008139214922448896228399482142720239819456249856157381518163968590180693114889682651996416899194740203776
735184372088832415615395299328218198082532147266823162775470081315581017142067217267000849989632151086257437409288498601980854272278860377496780840667138384947266540410775079424
81125899906842624147774362773094408727923301285888030547731554500608070164320914243584011050880543993364481208690059499274240906517544624455680446176603994849280130134842831831040170801981216778244923990397355877376
9360287970189639685201657569612922883413587780058480640134410018839802675203528262994544820224064831832524761071616850917801887489064967977354392695209984052351388202062315520229037323384022630406012229738830594048717368321110468608364375289404334925824
1011529215046068469761815851369755783987213108177075424565657657348274704982474752016935662373815137075203556489098501178785792544587393207993001574461266081735899212677125025733267397982289920293167773931548966912011543481098554740572162754694353064199454723012946948663968153626963771415920784510976
113689348814741910323262949514151533844889649572742394332902850562385688227727270949683278280394972301078036480184937433122061296861184323640507963607269507072424778166702234541228032418141007847512126521344304894484888810925588480160069604566625735933952572976425437353486581761253385930644210751897612654377184388666245121995319084778138053812224
121180591620717411303424216933710306824327004161850715716055095039754249716257509289248958709763506961694759088296034304920577444874260677709004818123868445962007092396032271858026689430106385940483122119525261423878026035227318545846037458932727808179277957114620824246026248556447953197812066287616280758448464303208425062456691609786061224778137653148384174432398229504147653612273582215982104576
13377789318629571617095687437727210519691211571206833411874664966300631040388650300371569958348390401530310557713056710996787204418771735396451253003223049666063171179737115944550401631148160136580638315642880214088196017926208778928128021854836676829967146182246401721068388300359912761851904102677375438373744795451392044921351754288513348009984013605986348654693946753024025511224403727551150161920223223213532616072563916810926367308245083982675738624
141208925819614629174706176253874422119072126688296962499076342734616247087923201530684259924952451341352960652932504624237530025295872020567373895663482195796819968494902434364402540336360980489279420644332547631306768384013701644545147277361851400192015985251969338490255493300224014686450246829737922234469580810514163244889471467054222540857499330245489297085452779520232208833683706776691251609606530873447354253094441451520114290285328699429152725401693753749683698750476845056808551180810136214718004658176
Technical details
Node id
n-66d9deddf7ecf8ee · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-039ab5acdf7230f1

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:              1024,               2688,                1764;
  1:             32768,             129024,              169344,                74088;
  2:           1048576,            5505024,            10838016,              9483264,              3111696;
  3:          33554432,          220200960,           578027520,            758661120,            497871360,            130691232;
  4:        1073741824,         8455716864,         27745320960,          48554311680,          47795650560,          25092716544,           5489031744;
  5:       34359738368,       315680096256,       1242990379008,        2719041454080,        3568741908480,        2810384252928,        1229543110656,         230539333248;
  6:     1099511627776,     11544872091648,      53034256171008,      139214922448896,      228399482142720,      239819456249856,      157381518163968,       59018069311488,        9682651996416;
  7:    35184372088832,    415615395299328,    2181980825321472,     6682316277547008,    13155810171420672,    17267000849989632,    15108625743740928,     8498601980854272,     2788603774967808,      406671383849472;
  8:  1125899906842624,  14777436277309440,   87279233012858880,   305477315545006080,   701643209142435840,  1105088054399336448,  1208690059499274240,   906517544624455680,   446176603994849280,   130134842831831040,   17080198121677824;
  9: 36028797018963968, 520165756961292288, 3413587780058480640, 13441001883980267520, 35282629945448202240, 64831832524761071616, 85091780188748906496, 79773543926952099840, 52351388202062315520, 22903732338402263040, 6012229738830594048, 717368321110468608;
%F Formula
The row polynomials satisfy P(n,x) = (32+42*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 2688*x + 1764*x^2.
%t Mathematica
P[0, x_] := 1024 + 2688*x + 1764*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 42*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] = 1024 + 2688*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 42*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. [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.