Real-rooted atlas

A038334 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(36 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1296 + 1152\,x + 256\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0129611522562704
14665662208276484096140608
2167961629859841990656589824655367311616
36046617613436928011943936053084160117964801048576380204032
4217678233658047528966449725440382205952012740198402264924161677721619770609664
5783641640962437996216323250661621762407897497601070176665602853804441642278584322684354561028071702528
628211099074561003061300428815603175784448138694895861767705271992320273965226393660881161420877309411328429496729653459728531456
710155995666841640623982667366472220413630873674895243765350449930162510233622191183337881665751654334464125241246351361391569403904687194767362779905883635712
83656158440062976162495930669465603249918613389312038517553936465920299580975061401601597765200327475259176488901017601502894956216320250482492702720247390116249601099511627776144555105949057024
91316217038422671366434838854510837761429964189891297280190661891985506304016947723732045004801054525032216133632468677792096059392148786600665415680330636890367590404898324301742080435406604599296175921860444167516865509350965248
104738381338321616896252713671377152901126177445300330404249691517708153043025920915177081530430259206507925913105281843233744801030916276224128551622974919147523570878415969976320705358699450859520940478265934479367599824371187712281474976710656390877006486250192896
1117058172817957820825698558331837089631436826282221823223901716484283028741562413613056475892082395823734784038071366591665898782722256080983209831038976100270265920436935065633423421973478978355282526967835750563840146714609485778780161778358902857924608131730289100587008450359962737049620325604337285010030592
12614094221446481549721638210307112225518649344110385331657540387209216196240589613405132816384239849609527495162331136213199652913329033183232142133101942219355455488721945914627145932472322807567445772234181836883187183578436568350721848604079520812630016298764295680131334144331960328533479260162269814212194729984720575940379279361056931425538820521590784
13221073919720733357899776147382613147155571933184045852368534670622379212808830826532603230976737280117744353768043079689830401151278125731976779189452885279861165331613273292804873134923733235044188160216583774388143779741696074868465220592911515648019964924058824776404172840333179916817730109440597528591360262668288061284983729257709568038911100780481085440115292150460684697654960434128018667122720768
147958661109946400884391936565949234485077396223426561886497448283591320744755203912735448291893109692825605651728980866067825111859206028510912923805680119316484912120003123100924541665283118806351189270428280422401559403175594635214140211206160605138151645290430464019166327096471785348005888464638232641740250860748886044117155877824233472011766716876017480237056011206397024778552606720664082786653543858176184467440737095516162857942574656970690381479936
Technical details
Node id
n-0e58f5cb99dc6bc2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-12e1fe8c90bb4450

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:               1296,               1152,                 256;
  1:              46656,              62208,               27648,                4096;
  2:            1679616,            2985984,             1990656,              589824,               65536;
  3:           60466176,          134369280,           119439360,            53084160,            11796480,             1048576;
  4:         2176782336,         5804752896,          6449725440,          3822059520,          1274019840,           226492416,           16777216;
  5:        78364164096,       243799621632,        325066162176,        240789749760,        107017666560,         28538044416,         4227858432,          268435456;
  6:      2821109907456,     10030613004288,      15603175784448,      13869489586176,       7705271992320,       2739652263936,       608811614208,        77309411328,        4294967296;
  7:    101559956668416,    406239826673664,     722204136308736,     748952437653504,     499301625102336,     221911833378816,     65751654334464,     12524124635136,     1391569403904,      68719476736;
  8:   3656158440062976,  16249593066946560,   32499186133893120,   38517553936465920,   29958097506140160,   15977652003274752,   5917648890101760,   1502894956216320,   250482492702720,   24739011624960,   1099511627776;
  9: 131621703842267136, 643483885451083776, 1429964189891297280, 1906618919855063040, 1694772373204500480, 1054525032216133632, 468677792096059392, 148786600665415680, 33063689036759040, 4898324301742080, 435406604599296, 17592186044416;
%F Formula
The row polynomials satisfy P(n,x) = (36+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 1296 + 1152*x + 256*x^2.
%t Mathematica
P[0, x_] := 1296 + 1152*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(36 + 16*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] = 1296 + 1152*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (36 + 16*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.