Real-rooted atlas

A034870 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 16\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
11616436
2256512384128161296
3409612288153601024038407686446656
4655362621444587524587522867201146882867240962561679616
5104857652428801179648015728640137625608257536344064098304018432020480102460466176
61677721610066329627682406446137344051904512041523609624222105610380902432440320720896010813449830440962176782336
726843545618790481926106906624122138132481679399321616793993216125954949127197425664314887372810496245762624061444771020859637764587521638478364164096
8429496729634359738368128849018880300647710720488552529920586263035904537407782912383862702080215922769920959656755203358798643291603599361908408320293601280314572802097152655362821109907456
968719476736618475290624262851998515270093866270721314259992576018399639896064199329432207361708537990348811746198683648652566593536029365496709121067836243968311452237824718735933441283457024017112760321604321289437184262144101559956668416
101099511627776109951162777605222680231936015668040695808033294586478592053271338365747266589172957184066589172957184054103703027712036069135351808019838024443494490172838379520338148143923201040455827456026011395686405202279137288128561152095630131207969177604194304010485763656158440062976
1117592186044416193514046488576101594874406502433864958135500808042927557181440144772696029265922050946527081267223439388880928768219744270758707201709122105901056011109293688356864605961473910374427773234220892161068201316188160343350423060480915601128161282002877467852835344896491524909013401605167382528038755368961845493764194304131621703842267136
1228147497671065633776997205278721942177339303526471213169107795968186934568907964416373869137815928832591959468208553984761090744839569408808658916392042496718807925681815552539105944261361664343067419075411968185828185332514816857668547688529923369412151633510411231373838778368315882389215641674325268050739214452135454310422819161243648285239515545627165668147218522046464805306368167772164738381338321616896
134503599627370496585467951558164483659174697238528001463669878895411200420805090182430720092577119840134758401620099597202358272023144279960033689600274838324525400064002748383245254000640023361257584659005440169900055161156403201061875344757227520057177903179235328002654688361893068800106187534475722752036501964976029696010735872051773440026839680129433600565045897461760098883032055808014126147436544016052440268800139586437120087241523200348966092867108864170581728179578208256
14720575940379279361008806316530991104680944263658418995229507584758531489792922112023704109056002213068856889861734404241715309038901657606665552628489702604808748537824892734668809720597583214149632009234567704053442150407555555394225543577605351851737576426700803293447223123954892801764346726673547264008233618057809887232033449073359852666880118055553034774118403607253009395875840949277107735756800213587349240545280406833046172467206472343916380160844218771701760879394553856007035156430848405874409472150323855362684354566140942214464815497216
Technical details
Node id
n-d4fa80d8a6076549 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-90334921cba1a1bb

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:          16,           16,             4;
  2:         256,          512,           384,           128,             16;
  3:        4096,        12288,         15360,         10240,           3840,            768,             64;
  4:       65536,       262144,        458752,        458752,         286720,         114688,          28672,           4096,            256;
  5:     1048576,      5242880,      11796480,      15728640,       13762560,        8257536,        3440640,         983040,         184320,         20480,          1024;
  6:    16777216,    100663296,     276824064,     461373440,      519045120,      415236096,      242221056,      103809024,       32440320,       7208960,       1081344,         98304,         4096;
  7:   268435456,   1879048192,    6106906624,   12213813248,    16793993216,    16793993216,    12595494912,     7197425664,     3148873728,    1049624576,     262406144,      47710208,      5963776,      458752,       16384;
  8:  4294967296,  34359738368,  128849018880,  300647710720,   488552529920,   586263035904,   537407782912,   383862702080,   215922769920,   95965675520,   33587986432,    9160359936,   1908408320,   293601280,    31457280,    2097152,     65536;
  9: 68719476736, 618475290624, 2628519985152, 7009386627072, 13142599925760, 18399639896064, 19932943220736, 17085379903488, 11746198683648, 6525665935360, 2936549670912, 1067836243968, 311452237824, 71873593344, 12834570240, 1711276032, 160432128, 9437184, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (16+16*x+4*x^2)*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[(16 + 16*x + 4*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (16 + 16*x + 4*x^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.