Real-rooted atlas

A034870 → x ↦ x+1 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 24\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
19241649
2814328647682562401
37295832194403456034560184324096117649
4656169984326592870912145152015482881032192393216655365764801
559049787320472392016796160391910406270566469672960530841602654208078643201048576282475249
6531441850305662355744277136640831409920177367449627590492163153199104262766592015571353606228541441509949441677721613841287201
747829698928208877377809641268165121513166054440351094784807021895681229747650561434705592321275293859848501959065641221619712137405399042818572288268435456678223072849
8430467219183300489183300480571405363202476089907207923487703041936852549632368924295168055338644275206558654136320612141052723244519349288962473297182720101468602368028991029248051539607552429496729633232930569601
93874204899298091736105378373008749357319168374678659584013988003291136404097872855049236522808115216933625148211225086852071424030104222485708829191973319475222704868137369613972226546073666534412124160236566798663685914169966592927712935936687194767361628413597910449
1034867844019298091736011777582865609422066292480533917089907202278046250270727593487500902402024930000240640438734833385472077997303712972801143960454456934413866187326750720138661873267507201137738447323136075849229821542404045292257148928168553844047872052879637348352011751030521856016492674416640109951162777679792266297612001
1131381059609920511081864128871551460961145524901875207254991045209603482395701700608131557170953134084009361400476467210023403501191168020789281335803904036034754315393433652414188095117721664061785449588326465704395332911104056318053142495232040048393345774387223361562785035059210993676604722380840717320758231040114294233707315202285884674146304290271069732864175921860444163909821048582988049
12282429536481903774516739213857875923334413549923125038089484946187526656505863796668088322135869363709706247322980675576135682074844524746571776491814998458446643298362999691689328641669190297798364364824110526523754151936296744941830820331523108756533465736806427633391408584327168207250435564382453761300394889815733043267427883175630602242839068765289709568946356255096569856240344445738811392436989901343293445066549580791808281474976710656191581231380566414401
132541865828329881180153820721468633589701200156654249568128001201015913355648007045960025019801603288114678342574080125261511555907584003966614532603740160010577638753609973760023975981174849273856046498872581525864448077498120969209774080011127935318655762432001377744372785951539200146959399763834830848013471278311684859494401056570847975283097600704380565316855398400395441720879638118400184539469743831121920703007503786023321602130325769048555520049398858412720128008233143068786688008782019273372467245035996273704969387480337647754305649
1422876792454961854066918318544153732045297337921776459190102571521480382658418809600947444901388038144048424961626499727360202923648720570286080710232770521996001280210439339413924741120053311299318194267750401163155621487874932736021970717294770970951680360545104324446702796805150644347492095754240064096907435457191608320694383163884119575756806535370954203478360064053251170737954268119040373692426231258021888002242154557387548131328011388721561333577809920483157884420212391936016805491632007387545604668192120002052096009958809856004377804815321245932314427392151320947479648665672057594037927936459986536544739960976801
Technical details
Node id
n-fae42818526cffa5 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-7cfe8b61bc182db1

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:         9,         24,           16;
  2:        81,        432,          864,          768,           256;
  3:       729,       5832,        19440,        34560,         34560,          18432,           4096;
  4:      6561,      69984,       326592,       870912,       1451520,        1548288,        1032192,         393216,           65536;
  5:     59049,     787320,      4723920,     16796160,      39191040,       62705664,       69672960,       53084160,        26542080,         7864320,         1048576;
  6:    531441,    8503056,     62355744,    277136640,     831409920,     1773674496,     2759049216,     3153199104,      2627665920,      1557135360,       622854144,       150994944,        16777216;
  7:   4782969,   89282088,    773778096,   4126816512,   15131660544,    40351094784,    80702189568,   122974765056,    143470559232,    127529385984,     85019590656,     41221619712,     13740539904,      2818572288,      268435456;
  8:  43046721,  918330048,   9183300480,  57140536320,  247608990720,   792348770304,  1936852549632,  3689242951680,   5533864427520,   6558654136320,   6121410527232,   4451934928896,   2473297182720,   1014686023680,   289910292480,    51539607552,    4294967296;
  9: 387420489, 9298091736, 105378373008, 749357319168, 3746786595840, 13988003291136, 40409787285504, 92365228081152, 169336251482112, 250868520714240, 301042224857088, 291919733194752, 227048681373696, 139722265460736, 66534412124160, 23656679866368, 5914169966592, 927712935936, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (9+24*x+16*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[(9 + 24*x + 16*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] = (9 + 24*x + 16*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.