Real-rooted atlas

A038319 → x ↦ x+1 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(21 + 5\,x\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 441 + 235\,x + 25\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
044123525701
192618315195012519651
219448126249510202514250625573876
3408410177972754475250102312596875312517479751
48576612122314965517771912559208750908750062812515625555574901
518010885416230721635662504115030201106256686312507433125039375007812518423940076
63782285936117100420521523652988382514184918825042514434375676239375057246875024062500390625637079886651
77942800465814635223632395818744742465062892201961252452337750625523642284375630078750004209843750144140625195312522949515147251
816679880978201124487214675175276986988326125267349164183750131991948756250363492338343755831613046875550790625000298417968758496093759765625860257535597626
935027750054222133200669864455559209032711485750110095048522456256748415719925000233202450016250047915843456250060002340234375457430273437520528320312549414062504882812533513267621775401
10735582751138664188073129150295535302080348988286025442387338238271250331732568778153125141015205607100000360972026635000005755923225937500579044167968750364202050781251377255859375284179687502441406251355114414453945701
11154472377739119461232668049546461711598048564643883444501743634592151919512515812972011095634375814536230181736562525460333638740000005028177139700000006410131037578125052989543750000002798861914062509046044921875161865234375122070312556799230009941323026
123243919932521508681612660547707756422953155839528733617635256767327354966327467507355135820625169850004537515259390769906251705745130666608281254088795880169000000064242685027726562506702375406289062504636669146484375020869303710937505834130859375091430664062561035156252464697261870192715001
136812231858295168230116091370237027742430751008943281292809288075025940298242245587700625335397205420463601187502455013810351012649375010968057796095087187500314259825822794414062559632017267758203125076357020195371093750664322279714843750039044016552734375015160297851562500370336914062500512695312500030517578125110587231488966789681751
14143056869024198532832142178174209186888731455320818102319284750518625984297900785481621626250150483003343512870337187512970049554716976193062506821216381184000790312502308257914412403078125005208852182447222460937580211641962824023437508534539195743945312506288956656582031250031801849536132812501076504638671875002318237304687500285644531250001525878906255124536484159928623695276
Technical details
Node id
n-395eb038698327b7 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b36eb96605c9fbf5

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             441,              235,               25;
  1:            9261,             8315,             1950,               125;
  2:          194481,           262495,           102025,             14250,              625;
  3:         4084101,          7797275,          4475250,           1023125,            96875,             3125;
  4:        85766121,        223149655,        177719125,          59208750,          9087500,           628125,           15625;
  5:      1801088541,       6230721635,       6625041150,        3020110625,        668631250,         74331250,         3937500,          78125;
  6:     37822859361,     171004205215,     236529883825,      141849188250,      42514434375,       6762393750,       572468750,       24062500,        390625;
  7:    794280046581,    4635223632395,    8187447424650,     6289220196125,    2452337750625,     523642284375,     63007875000,     4209843750,     144140625,      1953125;
  8:  16679880978201,  124487214675175,  276986988326125,   267349164183750,  131991948756250,   36349233834375,   5831613046875,   550790625000,   29841796875,    849609375,    9765625;
  9: 350277500542221, 3320066986445555, 9209032711485750, 11009504852245625, 6748415719925000, 2332024500162500, 479158434562500, 60002340234375, 4574302734375, 205283203125, 4941406250, 48828125;
%F Formula
The row polynomials satisfy P(n,x) = (21+5*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 441 + 235*x + 25*x^2.
%t Mathematica
P[0, x_] := 441 + 235*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(21 + 5*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 441 + 235*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 5*x)*v[i - 1] + 5*x*deriv(v[i - 1], x)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.