Real-rooted atlas

A038314 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(46 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2116 + 2208\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0211622085764900
1973361523527948813824343000
2447745693442567312896254361633177624010000
32059629765372947205606553602925158407630848079626241680700000
49474296896296586685443868521984026911457280105305702402197684224191102976117649000000
543581765721615916818785282491328157696216637231104011302812057603538271600646153515827245864714248235430000000
6200476122319368367699018547215280146033868815944500209254410398587092992043402798301184113224691220481687821484032110075314176576480100000000
7922190162669056433028424209817690371149400309761100170514438553686100301129973764492189624172544156250073884262434937904719462445571180068864264180754022440353607000000000
84242074748277657622132563904057344051963410905178112072296919520247808066010230866313216041328144542387404817968758496690176053571453903175680104813714158387201215231468503040634033809653762824752490000000000
9195135438420772249611199077335453016064292149843533556940804572780159655673856047715966883363553280348534018974133780481818438359865045811267767889187517235201767857978804797440307453561531269120320821107684802561521681143169024197732674300000000000
1089762301673555234816561990062651824078848161266713630523431321628046384979221466316803292401714952085176320274843969248174066892816729632910758421463047481574966301902766082439644010750620467205657145532175351808088546625721005506568399679910293012483652034743605657613841287201000000000000
114129065876983540801536280058381221492332625928767044977368455630028816771738217574436857446421876180283792743727104020544586701301011500236814291886400905051478425674566363830808964235264291781423685774207877128457432570602150952960176502940603870976409625115042931776107315221839167766761832448876488338465357824968890104070000000000000
121899370303412428768706561387366134666469709316096470498080452107118811545698190903746526703056322561408826010276252696025497614700793150708723784613888115049685527285664401326086860105472434424709644288313178728089397649788108810893173150935570427412482841697343722322720194565391362549354604370329670322120208973100482565644584899716904386562103572012316858777667822307284900000000000000
1387371033956971723360501766837733092284743567343616024972590423996454767689728056459769654252854257385472088371813371874032750690304010143547273989019411383582728820475890425234270768332805916840969974691312068198403087047462595491119339929601252714912357590599152435203921542334336805353868492893001003976366925388185601617408764806381311098880194738179040233201336320145146468849863255654405048572829560461066244747561509943000000000000000
14401906756202069927458308096335504770394771417704326758413128447537186707649299742720319649157427154621026428518405420137886808274008709005312067869552669599257152530153472649187025535297242328549294084838661059890414228535771136028400836655878518297927352320131714025070740954425170329604810425263453147900745351168136897477853212114171409203229760321272437416124219392047775766591203878727843840534139005367496780808192037157496025564993447526412116574790945106558976332329305696010000000000000000
Technical details
Node id
n-a1d15ac1f4dec54a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4b5617cf7459dd0c

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:                2116,                 2208,                  576;
  1:               97336,               152352,                79488,                13824;
  2:             4477456,              9344256,              7312896,              2543616,               331776;
  3:           205962976,            537294720,            560655360,            292515840,             76308480,              7962624;
  4:          9474296896,          29658668544,          38685219840,          26911457280,          10530570240,           2197684224,            191102976;
  5:        435817657216,        1591681878528,        2491328157696,        2166372311040,        1130281205760,         353827160064,          61535158272,          4586471424;
  6:      20047612231936,       83676990185472,      152801460338688,      159445002092544,      103985870929920,       43402798301184,       11322469122048,       1687821484032,        110075314176;
  7:     922190162669056,     4330284242098176,     9037114940030976,    11001705144385536,     8610030112997376,     4492189624172544,     1562500738842624,     349379047194624,      45571180068864,      2641807540224;
  8:   42420747482776576,   221325639040573440,   519634109051781120,   722969195202478080,   660102308663132160,   413281445423874048,   179687584966901760,   53571453903175680,   10481371415838720,   1215231468503040,    63403380965376;
  9: 1951354384207722496, 11199077335453016064, 29214984353355694080, 45727801596556738560, 47715966883363553280, 34853401897413378048, 18184383598650458112, 6776788918751723520, 1767857978804797440, 307453561531269120, 32082110768480256, 1521681143169024;
%F Formula
The row polynomials satisfy P(n,x) = (46+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 2116 + 2208*x + 576*x^2.
%t Mathematica
P[0, x_] := 2116 + 2208*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(46 + 24*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] = 2116 + 2208*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (46 + 24*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.