Real-rooted atlas

A038226 → basis A007318 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(19 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 361 + 304\,x + 64\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
036130464729
168598664364851219683
2130321219488138624389124096531441
324760995212840438976018483203891203276814348907
44704588111885275212510816070236160221798403735552262144387420489
589387173926345693363327877056233535232098330624024841420834865152209715210460353203
6169835630415720779129684306218752709947105283736563712012586319872264975155231876710416777216282429536481
732268769777912228165389522059480486656202334925004812779047895045380651745281510358384642725458739228689039361342177287625597484987
86131066257801258150158223204891266155808054919479644160404669850009602044647663206471742023270401726123868160272545873920255013683201073741824205891132094649
91164902588982195395338306864881135860696182080143477140570368012082285521715207122189360168962998816572702729018997211136018987362549760266489298944022441204121685899345925559060566555523
1022133149190661611118306485422902425897623872951424363475422778265603443451373688832023197988201693184113955029762703364112662728278016108227966533632020253186719744025582972698624195850508697668719476736150094635296999121
114205298346225705923018475158288074458151937241990924889778429426231603294503609922349056071623788572727705640209846216268185616930461564744499253464615467614208125063427994419202106331418853376241875378241536169737107537925497558138884052555153018976267
1279900668578288412147099341477727906081289034608864132166421710056570343278592251379602393448488962116880862260618854413369773866909171712643357539460290969623702646190642298886653374369303101441400710393537495042144628353741619222575035302543361462350464942084398046511104109418989131512359209
131518112702987479829995880802293946094520282596048866367436480515613843545652866560651301697110298357760603311045744276373504423376172452123770880229196123432728657920965036309190436454403160352825418973184079840492431637217281528047702040903680214462835374161920208384941254246401253443255664640351843720888322954312706550833698643
142884414135676211676811943184259823974182272613637134681255004928012057431418298343956480164996429934608917299201667332344602363795865612870635642544562634752774173572483883466752036671379749236585267201372496084181954068480404525161653628567552929053002840869437441629917548843630592021116340713763635201905233748610252801069604911500492828147497671065679766443076872509863361
Technical details
Node id
n-36db20e01be4b1a5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-fa8b89dcb83e73ac

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:             361,             304,               64;
  1:            6859,            8664,             3648,              512;
  2:          130321,          219488,           138624,            38912,             4096;
  3:         2476099,         5212840,          4389760,          1848320,           389120,           32768;
  4:        47045881,       118852752,        125108160,         70236160,         22179840,         3735552,          262144;
  5:       893871739,      2634569336,       3327877056,       2335352320,        983306240,       248414208,        34865152,        2097152;
  6:     16983563041,     57207791296,      84306218752,      70994710528,      37365637120,     12586319872,      2649751552,      318767104,       16777216;
  7:    322687697779,   1222816538952,    2059480486656,    2023349250048,    1277904789504,    538065174528,    151035838464,    27254587392,     2868903936,     134217728;
  8:   6131066257801,  25815015822320,   48912661558080,   54919479644160,   40466985000960,  20446476632064,   7174202327040,  1726123868160,   272545873920,   25501368320,   1073741824;
  9: 116490258898219, 539533830686488, 1135860696182080, 1434771405703680, 1208228552171520, 712218936016896, 299881657270272, 90189972111360, 18987362549760, 2664892989440, 224412041216, 8589934592;
%F Formula
The row polynomials satisfy P(n,x) = (19+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 304*x + 64*x^2.
%t Mathematica
P[0, x_] := 361 + 304*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(19 + 8*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] = 361 + 304*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 8*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.