Real-rooted atlas

A038336 → x ↦ x/(x+1) → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 + 330\,x + 484\,x^{2} + 198\,n x + 484\,n x^{2} + 121\,n^{2} x^{2}\right)\,P_{n}(x) = \left(432 + 7128\,x + 24684\,x^{2} + 23958\,x^{3} + 2376\,n x + 13068\,n x^{2} + 15972\,n x^{3} + 1452\,n^{2} x^{2} + 2662\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 2112\,x + 4356\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0144211243566612
1172838016156816170368366928
22073660825637635848177664585640018426640
3248832912384075271680245329920351384000185530752866889024
42985984131383296135489024058879180801264982400013358214144555561529638940831040
5358318081839366144227621560321236462796803541950720005610449940484666716848641596389048321689834289408
64299816962522559283236419449651223740085698568500681728000179534398095362240024087347215325334863872444494575641671388501672192
751597803523405455032325619000803328427321542574081836147253248004847428748574728064086714449928275680826490884800541416929281207269217792002951812278092800
861917364224454060670976084285012049920732551215841280367229450649600011633828996579328241922601433497603310272330596352028803248501575680144872306135040003213750657762304119926775477195776
9743008370688599360085688321236180176732160120870950613811206924898212249600025594423792474521663867566778443366410923898690967961601267342934069329920956157220491264000424215086824624128841418354032312324801499787971973120
10891610044825678461683944652817800994544943104193393520982097920124648167820492800052651386087376158721532821602682640793631460828229987729408456243456264958771204589554658358067200030543486251372937216121164242980652974082172495444647317504189864950796385718272
1110699320537907210200018912804864252450468091920384301693892732072755221605682422218752000102670202870383509504341600242883559948288817981533979680964608142347958354667136614417899263167596462080001588261285071392735232945081095249093197824338909289364981530624554307713450842193927428225699900243263488
1212839184645488641318156290270167043534306553286885376460768854354438389763629754646932750336001916510453580492177408717360510055475891404819631556815512343150592398574283393067982520326014152424312411258880066706973972998494879744529245413339492190781442846838030665844857241693123695859741488578561399909786521259622400287945519694409573416960
1315407021574586368169477237320450048048936552276279951360691153281531657584640593959851316268236800034497188164448859193344143472102011095178280960441710028349027720888320102490530015360338362368018042457272937233776640002401451063027945815670784238160436002771485851648017081028183995069143449608381132627376733972070402519837615738267320320003504129741194548530380811071808307676877121716224
1418488425889503641621693086377017606144671129859788982190080102078023118521735577609503357621060291788800060213273887038008773836827546643586130274229944329423147271445924712284160245977272036864812069683204948788280577069835878400076846434016894266101465088914536074250642505670328328198893528317633188855808053639248815211097421250560241904411110873662750720006727929103093533178331136870283472441989826412544422688367991708227033169920
Technical details
Node id
n-997c2382668fddd8 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4e3b020099acf776

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          144,           2112,             4356;
  1:         1728,          38016,           156816,            170368;
  2:        20736,         608256,          3763584,           8177664,           5856400;
  3:       248832,        9123840,         75271680,         245329920,         351384000,          185530752;
  4:      2985984,      131383296,       1354890240,        5887918080,       12649824000,        13358214144,         5555615296;
  5:     35831808,     1839366144,      22762156032,      123646279680,      354195072000,       561044994048,       466671684864,        159638904832;
  6:    429981696,    25225592832,     364194496512,     2374008569856,     8500681728000,     17953439809536,     22400240873472,      15325334863872,       4444945756416;
  7:   5159780352,   340545503232,    5619000803328,    42732154257408,   183614725324800,    484742874857472,    806408671444992,     827568082649088,     480054141692928,    120726921779200;
  8:  61917364224,  4540606709760,   84285012049920,   732551215841280,  3672294506496000,  11633828996579328,  24192260143349760,   33102723305963520,   28803248501575680,  14487230613504000,   3213750657762304;
  9: 743008370688, 59936008568832, 1236180176732160, 12087095061381120, 69248982122496000, 255944237924745216, 638675667784433664, 1092389869096796160, 1267342934069329920, 956157220491264000, 424215086824624128, 84141835403231232;
%F Formula
The row polynomials satisfy (36+(198*n+330)*x+(121*n^2+484*n+484)*x^2)*P(n,x) = (432+(2376*n+7128)*x+(1452*n^2+13068*n+24684)*x^2+(2662*n^2+15972*n+23958)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 2112*x + 4356*x^2.
%t Mathematica
P[0, x_] := 144 + 2112*x + 4356*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((432 + (2376*n + 7128)*x + (1452*n^2 + 13068*n + 24684)*x^2 + (2662*n^2 + 15972*n + 23958)*x^3)*P[n - 1, x])/(36 + (198*n + 330)*x + (121*n^2 + 484*n + 484)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 2112*x + 4356*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((432 + (2376*n + 7128)*x + (1452*n^2 + 13068*n + 24684)*x^2 + (2662*n^2 + 15972*n + 23958)*x^3)*v[i - 1])/(36 + (198*n + 330)*x + (121*n^2 + 484*n + 484)*x^2)); 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. [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.