Real-rooted atlas

A038336 → x ↦ x/(x+1) → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(34 + 44\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1156 + 2992\,x + 1936\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01156299219366084
13930415259219747285184474552
2133633669175041342809611585024374809637015056
3454354242939939207609254409847270406371763201649162242887174368
4154480441611994951936388071974406696143872064991984640336429096967256313856225199600704
5525233501444757997601281847222598144398420560384051560307814404003506253824172700269772831927780966417565568854912
61785793904896184882192506888374075778252821674078484889635061009313792036298456701337623487236689100886843564228608140482236252161370114370683136
760716992766464707174386338816366066741163622411053780027293696214573377000407042776831937652326423956981422882816132870653269770244298756429316096618121839509504106868920913284608
82064377754059776267154768172441601555783649945395205368978870399795201215915803002306560188824571760358195220363434209450393601505867403724062720730788592983736320210161425433231360271973609384181768335775831236199424
970188843638032384999158832964931584646514538977308672025099976219119042560649646443318375219201177006497306232750081523184878866889441281407986022481998643209110497792530579456039300186556014264320101718129909683978241196683881290399744650190514836423555072
1023864206836931010563705970944088109875226377793190274193817611378655886000632627203313196860923713617920686026644144204231475210357657176294848200704114891659434531089285129292707748381191045120534482537161793994752020750498501575531560964882470235664830955525265409077677758873650714860157241037295616
1181138303245565435904136503263107245380403210599076900091994243072502936590161227962122241627147791698090465689603790297208896728378900486540120674174747006730248463685578343790244003848214753649568972883886085906032035637823642009603057240112565461649981441079025922081927641169922327310812333569422131223167799941782139043843955759092264800909058048
12275870231034922482073649981194799268308516864420430050370315771641856217634379015222281791078477452234884829106166824962004646079372047453729587238913718011339744690044928575530619327377736592261126517037895324718487882956856225424979272081071931392363811573395289936347791361712054463036658523989606455389997333538952246722561102787277228829818486784101938319743841411792896308549209196654470906527744
139379587855187364390502418207435248304883816857601649379428375854181056512092494611081469469761208320359096725375116764955279360102236950047974420140208947222051106873092521991025459203669007698212033070775664640474812760945086632688615424047791611232381268911141683203710878048631957350747471872218286944037173961808674816094162995467016218819428352028121075569335160371412992051988543069359120014376960448528606872902211888742424066838317339048730709164032
1431890598707637038927708166603229850051904530913689664090172074033191035338752038705437252553378115459481601627905155033862667797266432050560818932816440505703333888119958021389623319631178498048221771131980816221166884618240322872677442658910228258488320371409093005934432680872509440336452943075964133134437449728237496195112445270447838199808128061673835142057594422558720509928836990610908068288921601414088371486568064391053312024399956213885880326747586561973525870240769732310466561877213388752445800995314794496
Technical details
Node id
n-4113fe25dfc072d5 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-6f86be61a76d4bf4

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:              1156,               2992,                1936;
  1:             39304,             152592,              197472,                85184;
  2:           1336336,            6917504,            13428096,             11585024,              3748096;
  3:          45435424,          293993920,           760925440,            984727040,            637176320,             164916224;
  4:        1544804416,        11994951936,         38807197440,          66961438720,          64991984640,           33642909696,            7256313856;
  5:       52523350144,       475799760128,       1847222598144,        3984205603840,        5156030781440,         4003506253824,         1727002697728,          319277809664;
  6:     1785793904896,     18488219250688,      83740757782528,      216740784848896,      350610093137920,       362984567013376,       234872366891008,        86843564228608,       14048223625216;
  7:    60716992766464,    707174386338816,    3660667411636224,    11053780027293696,    21457337700040704,     27768319376523264,     23956981422882816,     13287065326977024,     4298756429316096,      618121839509504;
  8:  2064377754059776,  26715476817244160,  155578364994539520,   536897887039979520,  1215915803002306560,   1888245717603581952,   2036343420945039360,   1505867403724062720,   730788592983736320,   210161425433231360,    27197360938418176;
  9: 70188843638032384, 999158832964931584, 6465145389773086720, 25099976219119042560, 64964644331837521920, 117700649730623275008, 152318487886688944128, 140798602248199864320, 91104977925305794560, 39300186556014264320, 10171812990968397824, 1196683881290399744;
%F Formula
The row polynomials satisfy P(n,x) = (34+44*x)*P(n-1,x) for n >= 1, with P(0,x) = 1156 + 2992*x + 1936*x^2.
%t Mathematica
P[0, x_] := 1156 + 2992*x + 1936*x^2;
P[n_, x_] := P[n, x] = Expand[(34 + 44*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] = 1156 + 2992*x + 1936*x^2; for(n = 1, N, my(i = n + 1); v[i] = (34 + 44*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.