Real-rooted atlas

A038320 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(29 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 841 + 696\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
08416961441681
1243893027612528172868921
27072811170672726624200448207362825761
3205111494243686035120160145324803006720248832115856201
4594823321147680272815277269608428838402615846404329676829859844750104241
51724987630949965158964620257145764277635488017700560640439462195260615475235831808194754273881
650024641296116559881256642398327630272198482286643210266325171203398507642887031395123283129794564299816967984925229121
7145071459758695402661259978889423358785856863397946897925359021739366422175262369792611731375718410848438190081122252226565159780352327381934393961
8420707233300201174085751710428032415967559872803576934351434240259019384069376012861652174479364435052473958401048682358374401627265728512014963363020806191736422413422659310152401
9122005097657058295553335479562653211489659612888248014263025726344032011803883359732992068381117394315264282956347838545928363241808035840173032589131776023866564018176019751639187456743008370688550329031716248441
10353814783205469041175687340626163937639984015452851103045515036614186359040513468926148385152033995184076031016961641146817463566336582081629839294464150538352544645120276852142610841603436785218617344258566912999424891610044825622563490300366186081
1110260628712958602189551951061800531703961370361256884078713282079168803548257358082150864279532680025601602023049582961674248838747859768064409636574129074902335488113505917818662420482609331444107182080431889342472912896487398631003914243361369868992512106993205379072925103102315013629321
12297558232675799463481172378562377704516775246363889191244663132647674023038550840794368873250897490268090393672269039792298048860164485664538832292687872212129948634433545830476805671057294904524821187771326150318489643836768261000658944659612813958630604868235808340547993643439241383903232128391846454886437929227194915558802161
1386291887475981844409495356048188164390342658015514070613993406509768027818333514746797879584034533103673478783574656031437032309649651254169621680711937689414658048011534565956997324054528047729238442747547811840153611342114589809049603813798838707057328128717328935180010782720989419220937945907209448035000998952960558504532078755840154070215745863681555098314991537910888601
1425024647368034734878752116568042395388514126622085141806260637814728951680992900519295578016625152013352800087078462982200320132607118106158528926679041005985033908788840133427259467095600519537347788802768295829679357773086720101822375344528101998592029493377686001243337523266568125184705000636416114772629628801725235201461296080154504724480129573051442271354880714885801060807475218488425889503641663759030914653054346432641
Technical details
Node id
n-553f5f9b006f122d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-38fe1fb5c666b805

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:               841,               696,                144;
  1:             24389,             30276,              12528,               1728;
  2:            707281,           1170672,             726624,             200448,              20736;
  3:          20511149,          42436860,           35120160,           14532480,            3006720,            248832;
  4:         594823321,        1476802728,         1527726960,          842883840,          261584640,          43296768,           2985984;
  5:       17249876309,       49965158964,        62025714576,        42776354880,        17700560640,        4394621952,         606154752,         35831808;
  6:      500246412961,     1655988125664,      2398327630272,      1984822866432,      1026632517120,      339850764288,       70313951232,       8312979456,        429981696;
  7:    14507145975869,    54026612599788,     89423358785856,     86339794689792,     53590217393664,    22175262369792,     6117313757184,    1084843819008,     112225222656,      5159780352;
  8:   420707233300201,  1740857517104280,   3241596755987280,   3576934351434240,   2590193840693760,  1286165217447936,   443505247395840,  104868235837440,   16272657285120,   1496336302080,    61917364224;
  9: 12200509765705829, 55533354795626532, 114896596128882480, 142630257263440320, 118038833597329920, 68381117394315264, 28295634783854592, 8363241808035840, 1730325891317760, 238665640181760, 19751639187456, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = (29+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 841 + 696*x + 144*x^2.
%t Mathematica
P[0, x_] := 841 + 696*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(29 + 12*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] = 841 + 696*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (29 + 12*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

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.