Real-rooted atlas

A038321 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(32 + 14\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1024 + 896\,x + 196\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
010248961962116
1327684300818816274497336
2104857618350081204224351232384164477456
3335544327340032064225280280985606146560537824205962976
4107374182428185722883082813440179830784059006976010326220875295369474296896
53435973836810522669875213811004211210070523904044058542080115653672961686616064105413504435817657216
61099511627776384829069721658926951301125156108238848281974669312098691134259221588685619226985857024147578905620047612231936
7351843720888321385384650997762424423139246082474931954647041624174095237127105761666662420725138194432388596341145642502724812820661046784922190162669056
81125899906842624492581209243648096976925569843201131397464981504086622618412646404547687466663936165801105555456041450276388864068004359700480661153497088028925465497642420747482776576
936028797018963968173388585653764096379287531117608960497814884591861760435588024017879040266797664710950912116723978311041024364762432222003207979178204856320116363015487488010181763855155240495651696641951354384207722496
101152921504606846976605283789918594662414564641194916184064212401017425861017602090822515285819392014635757607000735744747033461190662553628013754794649845767660011076662067201489446598239846401954898660189798415550330251509765669391237529689762301673555234816
113689348814741910323220983171383844614963255080824882592114278488358823249158183321696642462928766763008076105939556403825868844395131407902231756819422869990957226393663731292157828399104154902446216944025602710792808796520448323446869231403008235846675481231367937147732541444129065876983540801536
12118059162071741130342472311236768941442334722056350795616772266393635986138923293514661888432958233920875098275843788384546807657109913624861273588425249783808124306367942126248919044758603147784520466432138792591810381846937630360879458521029017648301399138556182528528296553077958246435558421841785651211112006825558016189937030341242876870656
133777893186295716170956824792424035065637371904075926798607388514451456014394455569317405864755201889272293472909519749120181842458246767541275852813259345913826799884697607458382076527574935142403263042158480814034124801110340734483054775500802914644428018018785689657961678966267419033608452744849247331942408534021242028556288053337632762678476801555680955578122248737103395697172336050176
14120892581961462917470617684624807373024042229432322776751491927351385653248056692009626850090790420480806089511881774728426291208463939874758634648476057667887851078793215409651712424299069242457596310323202088346981427720981839872081213493722189149293772802487163245242042697121792593527592614578370904064108195134070365848862720145647295863954027315201365443398724569006080796508649255998586882177953337809371136401906756202069927458308096
Technical details
Node id
n-58b6050751882152 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-5dbcc0e32705bbae

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,                896,                196;
  1:             32768,              43008,              18816,               2744;
  2:           1048576,            1835008,            1204224,             351232,              38416;
  3:          33554432,           73400320,           64225280,           28098560,            6146560,             537824;
  4:        1073741824,         2818572288,         3082813440,         1798307840,          590069760,          103262208,            7529536;
  5:       34359738368,       105226698752,       138110042112,       100705239040,        44058542080,        11565367296,         1686616064,         105413504;
  6:     1099511627776,      3848290697216,      5892695130112,      5156108238848,      2819746693120,       986911342592,       215886856192,       26985857024,       1475789056;
  7:    35184372088832,    138538465099776,    242442313924608,    247493195464704,    162417409523712,     71057616666624,     20725138194432,     3885963411456,     425027248128,      20661046784;
  8:  1125899906842624,   4925812092436480,   9697692556984320,  11313974649815040,   8662261841264640,   4547687466663936,   1658011055554560,   414502763888640,   68004359700480,    6611534970880,    289254654976;
  9: 36028797018963968, 173388585653764096, 379287531117608960, 497814884591861760, 435588024017879040, 266797664710950912, 116723978311041024, 36476243222200320, 7979178204856320, 1163630154874880, 101817638551552, 4049565169664;
%F Formula
The row polynomials satisfy P(n,x) = (32+14*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 896*x + 196*x^2.
%t Mathematica
P[0, x_] := 1024 + 896*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 14*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 + 896*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 14*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.