Real-rooted atlas

A038284 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(14 + 52\,x + 38\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 1064\,x + 1444\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0196106414442704
12744223446064854872140608
2384164170881698144307283220851367311616
353782472990403962336010754912014595952079235168380204032
47529536122623872832090560301137536061302998406655754112301093638419770609664
5105413504200285657616308974976737786963202002564614403261319514882950717656321144155825921028071702528
614757890563204570521630443419955216526427975685607180920320121755928555521652401887539212814545250304434779213849653459728531456
7206610467845047198571525479815591936347054987489281413009591920643835311749498886940087927664648073163507691525478218094504961652161012628482779905883635712
8289254654976785119777792095896772858880694109974978560329702238114816010738872898596864242903077468262403767476303589376038347526661534720231302541767987206278211847988224144555105949057024
94049565169664120908445779968164090033558528013361617018337280725344923852595202756310710639861767481414786022481921450478376881909760196850636862544896017810295716135014409668446245901864962385720502235525127516865509350965248
1056693912375296184660171736678427567125637832704249416851008962560152322434009044992066151457055356682242094796140086294937648736073463232167936826772674822688563209973765601035608064081214948465575665664400801044375568220169065737908494995456390877006486250192896
117937147732541442800679271339622445611062418959564845393866883631185923080298109960687616015049456480093645209654464699642243668377614783275617180424273923009452536354586370048453806334847120166912049270402069115903836163647289503817670803456164996429934608917299234449804052280982732820325604337285010030592
1211112006825558016422256259371204608744980686176339558480883617356288294912603738429552294772736327743718899817162342413343851412349698752512413931717281051879669769830878285424982142156817789208326007110542950424142497013866792879718423828958091608782582579216169650133591673895321667521615942470726156288130909255398667734384641056931425538820521590784
131555680955578122246333843890568069120120343033920793313280141546330373504516096011525915473271082024960688261809689616040919043113565329548263042252801086570757862761184133120294926348562749464264704062262229141024886900326401013984874582405300948172812510202998094610855854080113187550935141717267251207089769673959426246410240274909436337202242207744049745517051493739066163254960434128018667122720768
14217795333780937113694585402099149832192192548854273269301248024389521541280778158080215150422167726864465920140155132154976357423513669743863381881092146462722704353886236205613842432082579377597569849994117120199239133251279638081044480378554353177431312353984512560457094314638566342262784633850285236793616696606720529369468988970493065297920307898568697666511272673280111429958195345975508205568189032964795676208451420162857942574656970690381479936
Technical details
Node id
n-a72001243ef4423f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8f7225f9e9fdccc1

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:           196,            1064,             1444;
  1:          2744,           22344,            60648,             54872;
  2:         38416,          417088,          1698144,           3072832,           2085136;
  3:        537824,         7299040,         39623360,         107549120,         145959520,           79235168;
  4:       7529536,       122623872,        832090560,        3011375360,        6130299840,         6655754112,         3010936384;
  5:     105413504,      2002856576,      16308974976,       73778696320,      200256461440,       326131951488,       295071765632,        114415582592;
  6:    1475789056,     32045705216,     304434199552,     1652642797568,     5607180920320,     12175592855552,     16524018875392,      12814545250304,       4347792138496;
  7:   20661046784,    504719857152,    5479815591936,    34705498748928,   141300959192064,    383531174949888,    694008792766464,     807316350769152,     547821809450496,     165216101262848;
  8:  289254654976,   7851197777920,   95896772858880,   694109974978560,  3297022381148160,  10738872898596864,  24290307746826240,   37674763035893760,   38347526661534720,   23130254176798720,   6278211847988224;
  9: 4049565169664, 120908445779968, 1640900335585280, 13361617018337280, 72534492385259520, 275631071063986176, 748141478602248192, 1450478376881909760, 1968506368625448960, 1781029571613501440, 966844624590186496, 238572050223552512;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (14+52*x+38*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 1064*x + 1444*x^2.
%t Mathematica
P[0, x_] := 196 + 1064*x + 1444*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14 + 52*x + 38*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 196 + 1064*x + 1444*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14 + 52*x + 38*x^2)*v[i - 1])/(1 + x)); 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.