Real-rooted atlas

A038329 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(21 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 441 + 252\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
044125236729
192617938226821619683
219448122226495256181441296531441
3408410158344303333960952560136080777614348907
48576612114702763610501974040007520857304097977646656387420489
5180108854136021770823087580356147027636042007896072013536685843227993610460353203
637822859361864522499688645224996849401285696176433163204032758016576108288470292481679616282429536481
7794280046581204243440549423342107491361556140499424666917356896190547816256362948221444444263936317447424100776967625597484987
816679880978201476568027948606127303216482046684214982720233421074913608003008282752190547816256031109847552033331979520211631616060466176205891132094649
935027750054222111008721445612661572674492230380134800670762604077028954721488030811581888595288033091110272179659369612802566562423040244434516480139676866563627970565559060566555523
10735582751138664125219980039039912396313972042055763774418781352912024264120737268720110921694798942723697389826631424905483222848512161693432651520205324993843201759928518656914248581122176782336150094635296999121
111544723777391194615737545458881579989835792215225565681030416327309344976736011662363817840378520283501392032144198203238625536411994866396072968828461422772992140134308297984016015349519769612479493132288594261577728130606940164052555153018976267
1232439199325215086811297567973008603472424097690927302635916275402182026315839042163874287349624449612364995927712139712529928396901948844817303784388635064324325946097158766088239897327921459211771281897030656122299032696422487356451926016383984404070478364164096109418989131512359209
136812231858295168230129195279392693578129058390558785387156258072293072781907907748061965490955921063784038949737172293240092818547493891568209568068133651030250565760194667574372144473604325946097158766080741590759512931328963104882484326409172427452231680604775436410880246847116902404701849845762954312706550833698643
14143056869024198532832165397425839633615008961401373410849291750192018684978811323890002560173503374676578978595201189737426353684424652862319579475669184148482543656305129354455040817603812363006789120207645412663620771840415290825327241543686472064810294673408770483905987461120677348488780185604147031563960320157982154817536282110990745679766443076872509863361
Technical details
Node id
n-5a4e3575d52df992 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8d10fbbacee008e8

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:             441,              252,               36;
  1:            9261,             7938,             2268,              216;
  2:          194481,           222264,            95256,            18144,            1296;
  3:         4084101,          5834430,          3333960,           952560,          136080,            7776;
  4:        85766121,        147027636,        105019740,         40007520,         8573040,          979776,          46656;
  5:      1801088541,       3602177082,       3087580356,       1470276360,       420078960,        72013536,        6858432,         279936;
  6:     37822859361,      86452249968,      86452249968,      49401285696,     17643316320,      4032758016,      576108288,       47029248,       1679616;
  7:    794280046581,    2042434405494,    2334210749136,    1556140499424,    666917356896,    190547816256,    36294822144,     4444263936,     317447424,     10077696;
  8:  16679880978201,   47656802794860,   61273032164820,   46684214982720,  23342107491360,   8003008282752,  1905478162560,   311098475520,   33331979520,   2116316160,    60466176;
  9: 350277500542221, 1100872144561266, 1572674492230380, 1348006707626040, 770289547214880, 308115818885952, 88033091110272, 17965936961280, 2566562423040, 244434516480, 13967686656, 362797056;
%F Formula
The row polynomials satisfy P(n,x) = (21+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 441 + 252*x + 36*x^2.
%t Mathematica
P[0, x_] := 441 + 252*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(21 + 6*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] = 441 + 252*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 6*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.