Real-rooted atlas

A178822 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(30 + 20\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x) + \left(20\,x + 4\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1362460
275613443362436
3120964390426880358486464
416329610967041241856387072322562921184
51959552232404484354560023869440451584025804897388928
621555072440444928128810035211070259203571814404541644818923523261616512
72217093127698290688339346022404299234508821061877760447352012840874803212976128110804069376
8216166579212653833420882205109043214781010083841032412372992325908430848490043473923373793280843448323839635388160
920175547392198182445260818689795604480465018127974404452851859456019605084635136431474422579248335211724825978208256524812288136151810995200
101815799265282984842007347240437665440972813676877191577601748451990405120103176941233766431179399797145650539423334400438087057408018893242368031488737284949222149487616
111584697540608435215302459392840824267931648038161288090877952639079195135180804919245481076326419608578310537216431347837855334453619021840384037030755041280130993147084818320719872184603311009202176
12134699290951686174530070904832169229699853713408102075955483351449622080441240196218882174066962773049344111212182212549017631888115232669696053307950902542336524174116257792029520481275084887206626590721038174126087068145037290029056
131119040263290888556809597367091233146500406339174402637774108888662016072913858543060254720904955781159384514565823681578980186521621171387416690294784457092933729701068860007144491476582447833457589485568223871868547891256061402808320574988746752277816840589889699840
14911218500108288116188878149989171263442863293044359168662364346820528701440232019294241104068608035890192027532981698562862332066928456105984129305101937191944192035065974465692172288059017688762611138560624370512476661350441158039485756211216261544241266688349593158025216312136748236811207914574776211275776
Technical details
Node id
n-a8805f831bd9f0e3 · 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:          36,            24;
  2:         756,          1344,            336;
  3:       12096,         43904,          26880,           3584;
  4:      163296,       1096704,        1241856,         387072,          32256;
  5:     1959552,      23240448,       43545600,       23869440,        4515840,         258048;
  6:    21555072,     440444928,     1288100352,     1107025920,      357181440,       45416448,       1892352;
  7:   221709312,    7698290688,    33934602240,    42992345088,    21061877760,     4473520128,     408748032,     12976128;
  8:  2161665792,  126538334208,   822051090432,  1478101008384,  1032412372992,   325908430848,   49004347392,   3373793280,    84344832;
  9: 20175547392, 1981824452608, 18689795604480, 46501812797440, 44528518594560, 19605084635136, 4314744225792, 483352117248, 25978208256, 524812288;
%F Formula
The row polynomials satisfy n*P(n,x) = (6*n+30+(4*n+20)*x)*P(n-1,x) + (4*n+20)*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[Together[((6*n + 30 + (4*n + 20)*x)*P[n - 1, x] + (4*n + 20)*x*D[P[n - 1, x], x])/(n)]];
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 + 30 + (4*n + 20)*x)*v[i - 1] + (4*n + 20)*x*deriv(v[i - 1], x))/(n)); 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.