Real-rooted atlas

A094638 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 3\,n + 2\,n x\right)\,P_{n-1}(x) + \left(2\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
22838874
3280776404481508
436401853617924500838445492
5582405183368306244096087158438401892232
611065601676729641936864324959289711424116236846080103226520
724344320619114496233158534426668554881212434064242897792211576326451207119034256
86086080002577320345614280619654423198581612815057631950444903213184644643814442690355210321920603537020432
91704102400011965223029769603514241024215933484602881923337664070480089529067841684973424896181779033600945963417618579456061529153463008
105282717440006136345793177670577995717222421570649947066882566771776323904143384036360278441461004529497664783683243520544237079961622839533568037158912007410417032046368
11179612392960003447975622547456564008327339325442312336585284577283606015865950634242629419914379267841003697555182272642130075344627123225681388992768001727198754693125968533749760817496064001039061424179824704
126645658539520002107575827184988164876822197325405184265580887652162734085350961882182642086449936330734036027264244775335433093776006775528588923056896109932189064489676810533346207637913657985608599470081678514415206401961990553600167555968526222309184
1326582634158080000139272791716236738564541473393046182461443260651851425980796928839589541823862771302498804263141536115773446095941173573574718464214058321886696244851244879557219193387776057443737765695199232447823827196995993620548215229435084850547581544038405101175439360030752527313520959439488
14114305326879744000098958113494170708377645350550407962669236224426818910631628872101888139266204166691475783270420432030872007924946186241562895524420325389761536682285742169264117311360179764747903521293920256295121786622542323507203054895619285762637824197514907070476025856766925360476087910416229501236150272014283291230208006366743750754176815723392
Technical details
Node id
n-30d429ec30d4fab7 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5df92108b28d2015

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:           4,             2;
  2:          28,            38,             8;
  3:         280,           776,           404,             48;
  4:        3640,         18536,         17924,           5008,            384;
  5:       58240,        518336,        830624,         409608,          71584,          3840;
  6:     1106560,      16767296,      41936864,       32495928,        9711424,       1162368,         46080;
  7:    24344320,     619114496,    2331585344,     2666855488,     1212434064,     242897792,      21157632,       645120;
  8:   608608000,   25773203456,  142806196544,   231985816128,   150576319504,   44903213184,    6446438144,    426903552,   10321920;
  9: 17041024000, 1196522302976, 9603514241024, 21593348460288, 19233376640704, 8008952906784, 1684973424896, 181779033600, 9459634176, 185794560;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+1+2*n*x)*P(n-1,x) + 2*n*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[(3*n + 1 + 2*n*x)*P[n - 1, x] + 2*n*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] = (3*n + 1 + 2*n*x)*v[i - 1] + 2*n*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.