Real-rooted atlas

A038320 → x ↦ x+1 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(23 + 6\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 529 + 312\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
052931236877
11216712222313221627737
2279841427440182952276481296919177
364363431407480689679602231280226800777631944965
4148035889446787432398327436145290240240472801772928466561164307861
534048254471384505086216622184852834686445620019636002382488641339027227993644472808289
6783109852814219354276806648467749924419545518081441734759362463897139222195192329853747216796161778179923409
7180115266146312705993314406258012496908002210921727408094278806798402170906339392278785463040197220510727104775681007769674315628029357
84142651121364937928072208655297927969906443610612854063820805757656955969601716253200668162947378035571229546460979201687762137605038848000604661763241265656392061
99528097579139271124770000779190236550473799774572493883798560518963342870013134960012550736137123008276870496138790436889467849907229710987764480140064857856035251780608362797056147287546580812585
10219146244320203213319001587734487201346752783038861432224422641689677516818674791853097425768657612160556170242386580095446243844059038387076710443228482022725122861158671360001132978926182424379962163221767823366961903317756931033
115040363619364673839756592351022134806491277483590008424401000937997910009504801012368802109385520805709021956280615772819275389859466540800407032300886760806455046452558595097647968010983107072265706771154739289676901896192166959205171213060694016341756170468874501141
12115928363245387498092859653963512607136721778010746671162294236439861228158504402432053586961686601176506403633202918094357672064147978931908536285356838022333473319376896063504919369808167680699632399921495040050834468595707596823923650538045440696672656289792113366824058887836416409617397127221298258826533
13266635235464391245607836255351213379319534263946168511598244580292191011168025013778762536278250393616535793726000224711931670961662523232109106787334602554392896335932526721697721038086790189283655542519040919746464158400186880841705529297270077445179273013628499968209725805575802880532504086073344076405059993600470184984576917048391279958970257841
146132610415680998648961244113863264666350138560228829121897874131948227282151349000951543967457281422383550812726497769721613579234740260182876087296778557204276732068432025628382021476986505131048966836629551725621514746881115616134421300695040001250463467811774276403296597861579129854361650999589611472273408179618396038889472040125586583715840511561263218688282110990745649989265049929804196521633
Technical details
Node id
n-428b25ab66142992 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-ef12d04f3c0ae80e

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:             529,               312,                36;
  1:           12167,             12222,              3132,               216;
  2:          279841,            427440,            182952,             27648,              1296;
  3:         6436343,          14074806,           8967960,           2231280,            226800,              7776;
  4:       148035889,         446787432,         398327436,         145290240,          24047280,           1772928,            46656;
  5:      3404825447,       13845050862,       16622184852,        8346864456,        2001963600,         238248864,         13390272,          279936;
  6:     78310985281,      421935427680,      664846774992,      441954551808,      144173475936,       24638971392,       2219519232,        98537472,        1679616;
  7:   1801152661463,    12705993314406,    25801249690800,    22109217274080,     9427880679840,     2170906339392,     278785463040,     19722051072,      710477568,      10077696;
  8:  41426511213649,   379280722086552,   979279699064436,  1061285406382080,   575765695596960,   171625320066816,   29473780355712,   2954646097920,   168776213760,    5038848000,    60466176;
  9: 952809757913927, 11247700007791902, 36550473799774572, 49388379856051896, 33428700131349600, 12550736137123008, 2768704961387904, 368894678499072, 29710987764480, 1400648578560, 35251780608, 362797056;
%F Formula
The row polynomials satisfy P(n,x) = (23+6*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 529 + 312*x + 36*x^2.
%t Mathematica
P[0, x_] := 529 + 312*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(23 + 6*x)*P[n - 1, x] + 6*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] = 529 + 312*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 6*x)*v[i - 1] + 6*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.