Real-rooted atlas

A109983 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(1 + 8\,x + 16\,x^{2} - n - 8\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 20\,x - 32\,x^{2} + 6\,n + 40\,n x + 64\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 20\,x + 32\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
13203255
213176880192015364525
3631284108004793611827215360081920413875
43218736103296692736288025675980801241088011468800458752039750625
516835730087312078387204590208018312704050388992094388224011514675208257536002642411523927034375
689893674406853296771065605827084803115376640120795750403421716480070257868800101953044480992225525765813305344015502147584395147828125
748639232038851200240692489280641268812843003663360215326863360817675059200236586205184051895310745608493283147776100550804766728138627481600403055837184092127048499240277377421875
82657291449171236923968058338990086396617113651603636224031680770867201509628706816056420290723840165911388487680382549754904576684257043283968930632674836480930252872220672644428704251904276381145497600552762290995204144960687890625
91462563897499082593181088468711214085938324807685599657082880407251353600002335504269312001070531825172480395101844406272011767737904791552282146167188357125404677259866931281583741171924992949009939295109128207598750793728049693329960468480187939178938368003341140958904320429720307521484375
10809745355219608017840070000363045352320521872248576056324446519296473537411481600317542399082496017247944018165760766279475684966402799518329392332808424599635099648002084427567149875200421686229566750720069064826868989952009016078090308157440916080047079555072069813140336305766403753771867329003520126963356438364160020314137030138265644812510520517578125
1145046719337952538012072203985627313851523204392219710208053393340934348850950789867929603911799927943987224569184688570368012766419370468966405527843444322467840200266363773648896006078175254650290176015432737094673367040032643660961189724160057100007935441895424081661749289606447104093898799045117214720084683723459149168640057679452829948837888027891310142379838668885319375526580715520124100909856844677124695547234028857421875
12251595969205935007688059414694402006870274816035695220006476848261002426081285153438312303001644577826789883904031787073298435276801891481579784083865694720610026012016640401405418368642580480144402914430775459840044141425896431026176001145301434412929384448025141306317853803151360464295219630997871001607151837515320655872000090760730963124619837440932388418477801183641607558900545982963698892846538875370565560958976204456248989151605555205708641853414855147520761152247121980686336493961984221113525390625
131409933619125033211732532014534480014458334114764828183124294607364194926328025804849560453835392614447692602353334681603806444653286070681625524389905166578483214514119179119781478407044727397679271772160293129306895475015680001048205777070573014220803223744555894072514969608519691113016458634854401929845194991119955394560373018377091414948315136061117511166756154402406408409190395360312589025280959045147099242884287692889028806016349865288663046554782540303798296903680368188370739080107497881614819634251464963962961923805761235609903431680004684013828442958069760052137620317986341552734375
147923848253756798058128347854710547681024511665527168217185540765473283529044587306434564570391732401238016484301550763808194564277442664671819202563191499839061372174336203160051960885962342401111366134355485681254405251642606003109547212802151074187609152757432320765239263589352821751808023658613645717387702435840635222717071615258630553601478124858591860897087488002970330131521367396528947205127744114359135531905843207549131870563721740141199369384041202576912615536066569717010451633635448544296968227159534617625087033999365547665764671027452166799362864977058167135303368704001063973741130817925545984002529367467359197357670400028907056769819398373376005518844894809846807861328125
Technical details
Node id
n-da7295b3362987cc · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-487455b1f16acaff

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       3,       20,         32;
  2:      13,      176,        880,        1920,         1536;
  3:      63,     1284,      10800,       47936,       118272,        153600,          81920;
  4:     321,     8736,     103296,      692736,      2880256,       7598080,       12410880,        11468800,          4587520;
  5:    1683,    57300,     873120,     7838720,     45902080,     183127040,      503889920,       943882240,       1151467520,        825753600,         264241152;
  6:    8989,   367440,    6853296,    77106560,    582708480,    3115376640,    12079575040,     34217164800,      70257868800,     101953044480,       99222552576,       58133053440,       15502147584;
  7:   48639,  2320388,   51200240,   692489280,   6412688128,   43003663360,   215326863360,    817675059200,    2365862051840,    5189531074560,     8493283147776,    10055080476672,     8138627481600,     4030558371840,      921270484992;
  8:  265729, 14491712,  369239680,  5833899008,  63966171136,  516036362240,  3168077086720,  15096287068160,   56420290723840,  165911388487680,   382549754904576,   684257043283968,   930632674836480,   930252872220672,   644428704251904,   276381145497600,    55276229099520;
  9: 1462563, 89749908, 2593181088, 46871121408, 593832480768, 5599657082880, 40725135360000, 233550426931200, 1070531825172480, 3951018444062720, 11767737904791552, 28214616718835712, 54046772598669312, 81583741171924992, 94900993929510912, 82075987507937280, 49693329960468480, 18793917893836800, 3341140958904320;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n-1+(8*n-8)*x+(16*n-16)*x^2)*P(n-2,x) + (6*n-3+(40*n-20)*x+(64*n-32)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 20*x + 32*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 20*x + 32*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (8*n - 8)*x + (16*n - 16)*x^2)*P[n - 2, x] + (6*n - 3 + (40*n - 20)*x + (64*n - 32)*x^2)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 20*x + 32*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (8*n - 8)*x + (16*n - 16)*x^2)*v[i - 2] + (6*n - 3 + (40*n - 20)*x + (64*n - 32)*x^2)*v[i - 1])/(n)); 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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.