Real-rooted atlas

A272867 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 48\,x + 124\,x^{2} + 128\,x^{3} + 46\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
16364688
2364321848331221167744
321638882829610627221693622852897336681472
412963110431968018351366429024140693761879008014016384447745659969536
57776233280309744023950080119344320400277376914973120140773248013957982408059420802059629765277319168
64665616796162734041626593920017207510407799763456253896629765979818649610114192224011984008320094456411776444880028169474296896464404086784
727993611757312226654848265715251221155976576120978821376512284010112163160610508839275107441927110866278656953350988889691800026680326003408824448238752281779243581765721640867559636992
81679616806215681796069376246433259522330194728961609717100544840211655270433796886777856105867948713472259109465296896493857739597824725386220826624805042003885056652727977316352364722533308416125515485278208200476122319363596345248055296
9100776965441955841375605504021622704537623678957629441918059662131211908141068902457930450318950422383929777080326920559737247744171610128290949123405023135413862453661611994568704662656461617848326271847456195174443908591067324416214160796755942406495426363147264922190162669056316478381828866048
106046617636279705601025909452801817613250560226243267430402102742712811521513913147596800864506524852224039763064872765440148753512767692800455072304490057728114044359788564480023371845908547686403895722551065559040523030826094612480055695363540763115524594255586327024640282974661559719936012245081551266508803319884585608601604242074748277657627850097600940212224
113627970562394460569674893405593614765840179200205899220961280215973159996211217694071681882112116043881283551232619440742972846080272166441345773568099146612707244974083006792613626981580876012403075554480128159973386079904686080279138352583356231680400911552867534077952468663022268231104512438570353088866414592320553684063302215680176242568653395763200685334841289135656961679861600317952409619513543842077224962450808588882738675712
12217678233615672832819253722988052481166581189509121801416538791936210499902026219521933861318541475841432597521847320576870815148141149798443967871766812229632185962160685576388608662531925582756642816199459417781236713062450794114294678009282561093044255585221217484819813225769881644367872300851582556750000906243794522235209608583577639270353839917635026944327716242802871094149122150140284320763079065610675155282716170616832376899857234973322444884298509397773611827289762301673555234816215671155821681003462656
13130606940161018734133248379761446338569005609737912321525531726002585619651721008398336020009815992037048321652578542715384627211271818174026331750464320016993666405171230991913158636275056641269235143945065240985644372973779389955801088132758327554858827841536340192798975322994475008746028212385488347201536139658743481899095783014422221454167314321591992322985567407476014904418304335584271211517746590515231152231677857929349201922345596150944978765250560139598411239943155792281663179900217340066116403220425997152132755999129642008757183223849893888412906587698354080153618979061712307928304713728
14783641640966582589784064265132088524800681736881969561612569900126979686417697204866592276481978225375220681932818022984179811968614413632825193824887930888677241691918540668928469353903729537835008002173122038248810510417928657252950262900868218882977825187601921386741760886161337967879373599539222829993104948063965020160508854090076563839920865289792731792360473140835123216215366144886123121786880022983417211147348067765452827683777738675783198534860828059037570325658168026726423611763182656229990288588816194364500288896325373132888185593077744240826302464366681656178435656725954561093304834383902760058880020810492019997045639741441899370303412428768706561670157430683097690814808064
Technical details
Node id
n-8c232b5bef38e52d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-42aa5bf2604973b0

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:        6,        36,          46;
  2:       36,       432,        1848,         3312,          2116;
  3:      216,      3888,       28296,       106272,        216936,         228528,           97336;
  4:     1296,     31104,      319680,      1835136,       6429024,       14069376,        18790080,        14016384,          4477456;
  5:     7776,    233280,     3097440,     23950080,     119344320,      400277376,       914973120,      1407732480,       1395798240,        805942080,         205962976;
  6:    46656,   1679616,    27340416,    265939200,    1720751040,     7799763456,     25389662976,     59798186496,     101141922240,     119840083200,       94456411776,       44488002816,        9474296896;
  7:   279936,  11757312,   226654848,   2657152512,   21155976576,   120978821376,    512284010112,   1631606105088,    3927510744192,    7110866278656,     9533509888896,     9180002668032,     6003408824448,     2387522817792,      435817657216;
  8:  1679616,  80621568,  1796069376,  24643325952,  233019472896,  1609717100544,   8402116552704,  33796886777856,  105867948713472,  259109465296896,   493857739597824,   725386220826624,   805042003885056,   652727977316352,   364722533308416,   125515485278208,    20047612231936;
  9: 10077696, 544195584, 13756055040, 216227045376, 2367895762944, 19180596621312, 119081410689024, 579304503189504, 2238392977708032, 6920559737247744, 17161012829094912, 34050231354138624, 53661611994568704, 66265646161784832, 62718474561951744, 43908591067324416, 21416079675594240, 6495426363147264, 922190162669056;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+48*x+124*x^2+128*x^3+46*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((6 + 48*x + 124*x^2 + 128*x^3 + 46*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((6 + 48*x + 124*x^2 + 128*x^3 + 46*x^4)*v[i - 1])/(1 + 2*x + 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.