Real-rooted atlas

A109983 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(1 + 6\,x + 9\,x^{2} - n - 6\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 15\,x - 18\,x^{2} + 6\,n + 30\,n x + 36\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 15\,x + 18\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
13151836
2131324958104861936
363963607520223374223645014580115776
43216552581042922489113311803060220887015309004592707270656
516834297549113033069601452370543456905896815801259930701152767706200145014880348469656576
6898927558038549793252933018437260573929348021499048354567440150703372005076551123605587570674245525742049105148430900391936
7486391740291288001352921439152029014603102049708953832355551510914644497523685334659038965431267047828662547442467769174625780202910095755039380164151496082059470913536
826572910868784207697320246117614420239296336122457847680563849657280201511107288056483997717151245740455476021542721249924288996893040242947904821323022100263088904114823971507963693408661800554011299270138581681176576
9146256367312431145866436219773754344187892308368132882487416072481991400003117521873160010717406166010529666098035616566268268159869811916481752137161712004594600762193820647860239416909409762696881096819258932540498056158043730141272881313850188363841751809394264067014656
1080974534141470601003503937515315975801016512364115101336605517987284279485588400423867936275280172674195408913557535854034069801576507166216734535581411160445150660270613857327001001811108859704001230593494716085501204860080243734209181499993498547052478166312051480211626776208147305368369489926300644204338791156640572085974859776
114504671925346440356790614741911523031111351389725767683012670490085006690681459506154052216225844808962459692699305880595856259334160105311292293113515285845827711663498425192534419030344776036663960967944453005816430539782369650763053729724993707081846270198971892907058322787163011650477422284208068485024388502592735180908844925571602571882029243667192141402213720364209608843888128613297422336
1225159596915445125576453342076560846648397188011294190705174311452562099157969172013011886934595042158138892203182296568466703051420213010211567792533405046599005086016953412448823434640457415755081210977751048679509155088707002040692141820704357703359748667230798499804653443298556210544105375998225517247900005116826908740212265603942401712534116488802397087565663585431781106884778997740062323647104300035330498076373352565231503607637335256523150363018885817056557531136
13140993361993774908799299258175645060996097046664891731667134061995475681357686188206960073245156636619392985491630381074270175322109119164908084998846599817341063888401009102975350199500775233809285267113795169461252490250139112006395685574405129058663402604011385256000738558771170193420582641785354813102803959302812628001327034456281082657558555860355564333047142532570403041335420457826774367821174690941106541316478116924651150217152533204925826662615547877188148698917444505737509228640007211961813850026436929734118597400208351593435165282533376
147923848253567598543596195668274683074322158588942746871886250782553837458822964319928134315363575445596464645333987112558428227254072750942812396327641763495960521144065657447658080135469386998659780554717016635287989240280037405511036373974619396282401363499341655784575827953161609610089377913858706366580509264897335696101111098595271497421916600167458602233883352995603021681551140338941434182602393989711651878318664686223190876227953094102699217333268499668925269369541100674682143342658427348556647943905215466303636215601974291648534662800600514858910503939941001070695654231803194700091773913219868845260014420701469216635046854656
Technical details
Node id
n-9906787568ecda50 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-c986abc9710a95e7

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,       15,         18;
  2:      13,      132,        495,         810,          486;
  3:      63,      963,       6075,       20223,        37422,         36450,         14580;
  4:     321,     6552,      58104,      292248,       911331,       1803060,       2208870,        1530900,          459270;
  5:    1683,    42975,     491130,     3306960,     14523705,      43456905,      89681580,      125993070,       115276770,        62001450,        14880348;
  6:    8989,   275580,    3854979,    32529330,    184372605,     739293480,    2149904835,     4567440150,      7033720050,      7655112360,      5587570674,       2455257420,        491051484;
  7:   48639,  1740291,   28800135,   292143915,   2029014603,   10204970895,   38323555515,   109146444975,    236853346590,    389654312670,    478286625474,     424677691746,     257802029100,      95755039380,      16415149608;
  8:  265729, 10868784,  207697320,  2461176144,  20239296336,  122457847680,  563849657280,  2015111072880,   5648399771715,  12457404554760,  21542721249924,   28899689304024,   29479048213230,   22100263088904,   11482397150796,    3693408661800,    554011299270;
  9: 1462563, 67312431, 1458664362, 19773754344, 187892308368, 1328824874160, 7248199140000, 31175218731600, 107174061660105, 296660980356165, 662682681598698, 1191648175213716, 1712004594600762, 1938206478602394, 1690940976269688, 1096819258932540, 498056158043730, 141272881313850, 18836384175180;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n-1+(6*n-6)*x+(9*n-9)*x^2)*P(n-2,x) + (6*n-3+(30*n-15)*x+(36*n-18)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 15*x + 18*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 15*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (6*n - 6)*x + (9*n - 9)*x^2)*P[n - 2, x] + (6*n - 3 + (30*n - 15)*x + (36*n - 18)*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 + 15*x + 18*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (6*n - 6)*x + (9*n - 9)*x^2)*v[i - 2] + (6*n - 3 + (30*n - 15)*x + (36*n - 18)*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.