Real-rooted atlas

A204420 → basis A007318 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x - 2\,n + 2\,n x + 4\,n^{2}\right)\,P_{n-1}(x) + \left(-x + 2\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
22421348
3720855225151815
44032058905217352730105123795
53628800619447528775255150253780094513254570
647900160092572672551127807511663190012318075592515103952045559285
78717829120018674373217511838028702532422264875439864425030648618010405395135135429440246235
820922789888000489273260726253476383347937511016049794750180592724812516252281045080310730502027025002027025117606685095900
964023737057280001615921374955387512651467926222125452371494587512586268963730950094247182104075603956558205022267680435043418875503445942540699973765526075
102432902008176640000656917138648082812555953386544023718752217241321945650000479337058584759375611585146083161254774241857510125228984946690500654074345925010148300662565472907517360158813883961075
11112400072777760768000032240007218419494243752958003280934892815625128155380576391605937530828010174813726875044742956013195189750409158301899059212523971091154222712589293522598280002034210499571252571121077525137493105758944922023368411430350
1262044840173323943936000018796524317980058956556251843037978601430740504375863878988840894407631250228008123058370470440625369339940010083487445003852280271406997930500265020289366391559000120853542287046318753597716788935018756689300831638425702039797959503162341432255476089670479451335537225
13403291461126605635584000000128427660150698496555154687513371178958159516657444531256723882163746842140452656251926030670148002130212500003432404842724073203800312540051740671508070630125003149487455124479844375001689805832855872606875061693407474942993750015014662720812129375232171202102214375205552193096250079058535806253928338943320231339886861875
1430488834461171386050150400000010164774484303875801876292031251117740963851609443322898796875598891120292641382130987562500184563531742903857829582171875357830099580710979974345343754603499101179940648871559375405766264322335135994775000249285288241951773186375001072564715545766111418750320622639697878530512506498721692755592300008488586142199288125642508720497393752134580466768753263379718067058428234040645000
Technical details
Node id
n-20aa41dfc4e18a9a · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-7d048716a74a4e37

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:                1;
  1:                2,                 1;
  2:               24,                21,                 3;
  3:              720,               855,               225,               15;
  4:            40320,             58905,             21735,             2730,             105;
  5:          3628800,           6194475,           2877525,           515025,           37800,            945;
  6:        479001600,         925726725,         511278075,        116631900,        12318075,         592515,         10395;
  7:      87178291200,      186743732175,      118380287025,      32422264875,      4398644250,      306486180,      10405395,       135135;
  8:   20922789888000,    48927326072625,    34763833479375,   11016049794750,   1805927248125,   162522810450,    8031073050,    202702500,    2027025;
  9: 6402373705728000, 16159213749553875, 12651467926222125, 4523714945875125, 862689637309500, 94247182104075, 6039565582050, 222676804350, 4341887550, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (4*n^2-2*n+(2*n-1)*x)*P(n-1,x) + (2*n-1)*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n^2 - 2*n + (2*n - 1)*x)*P[n - 1, x] + (2*n - 1)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4*n^2 - 2*n + (2*n - 1)*x)*v[i - 1] + (2*n - 1)*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

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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.