Real-rooted atlas

A156992 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 4\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x) + \left(4\,x + 4\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112820
221638496696
351841881611520153637056
415552010444801182720368640307202782080
55598720664012801244160006819840012902400737280278254080
62351462404804853760140520038401207664640038965248004954521602064384035581271040
711287019520391912980480172757975040021887012044801072241049600227742842880208089907206606028805640934440960
8609499054080356785009459202317839158476804167624661401602910969711820809189250301952013817169838080951268147200237817036801082616075878400
936569943244800359223005675520033876888256512000842886005391360008071183230566400035535929632358400782085107220480087611796357120047087773286400951268147200246787058810880000
102413616254156800396754383745843200537509891400007680018179725990035456000232409618416926720001371459649630371840041444617576513536006717855655526400005823187963084800025113479086080004185579847680065786583648573849600
11173780370299289600477263796297007104009220608279718133760004184825561719662182400700824077372856729600053945202752480083968002150306864577065779200473022675112742092800587994442581934080004060849568219136000143649100372377600200907832688640020243883090362970931200
12135548688833445888006213465931116655411200170296842431614471372800102719634449693524623360022219677904957926801408002187776373267712337510400111913473149795209052160032089197935783942553600053644104569410132377600527479496567370547200029706633957007687680087756541318397952001044720729980928007112715928348705003929600
131138608986200945459200870644303013799054540800337261352087389690920960002683901019893980924477440007418890747645325192724480009207807961372116809416704005925520641321317689196544002154161271382168836243456004650861447496519825489920061056492957031062306816004866992407608570150912002277867290821216174080057041751856958668800058504360878931968002826750400234521945440256000
14102474808758085091328000130664961991241914908672000713472705201601145261260800074488895649661823106416640000260926800791659616906772480000403617854979384506723598336000321895276627691996870344704000145415348688418516783595520000394348778790231760227532800006637075924790868820623360000702161433867177714253824000462859591268105422110720001828758564534094921728000393149305106422824960003510261652735918080001260431940846961714953977856000
Technical details
Node id
n-a67efd75592de231 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5162632e955115eb

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:             12,                8;
  2:            216,              384,                96;
  3:           5184,            18816,             11520,              1536;
  4:         155520,          1044480,           1182720,            368640,             30720;
  5:        5598720,         66401280,         124416000,          68198400,          12902400,            737280;
  6:      235146240,       4804853760,       14052003840,       12076646400,        3896524800,         495452160,         20643840;
  7:    11287019520,     391912980480,     1727579750400,     2188701204480,     1072241049600,      227742842880,      20808990720,       660602880;
  8:   609499054080,   35678500945920,   231783915847680,   416762466140160,   291096971182080,    91892503019520,   13817169838080,    951268147200,    23781703680;
  9: 36569943244800, 3592230056755200, 33876888256512000, 84288600539136000, 80711832305664000, 35535929632358400, 7820851072204800, 876117963571200, 47087773286400, 951268147200;
%F Formula
The row polynomials satisfy P(n,x) = (6*n+6+(4*n+4)*x)*P(n-1,x) + (4*n+4)*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[(6*n + 6 + (4*n + 4)*x)*P[n - 1, x] + (4*n + 4)*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] = (6*n + 6 + (4*n + 4)*x)*v[i - 1] + (4*n + 4)*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. [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.