Real-rooted atlas

A225471 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1628
260364100
384069614481688
4151201560045604801635776
533264040464014976022560144032911072
6864864011995200530208010214409408040326427065536
725945920040114368020426112047376000561792034944010752128918218240
8882161280014960090880856420992023035622403307046402661120011934722764825635008013056
93352212864006160468608003896169984001184851584001981953792019387468801128960003824640691205121481245379072
1014079294028800277765044480001915447564800064665216000001227947212800140453913600997387776043997184011658240168960102468855622529024
116476475253248001361430801561600101327679129600037456807449600079242192691200103213135872008593927372804634606592016027545603412992040550420483487394075052032
123238237626624000072089696731852800574398083331072002300236575436800053451833250816007777632006144007392497688576046849336934401984739328005526343680966451209584644096191116007619727360
13174864831837696000041017877695162368003475688276783923200149502156192829440037740609766379520060467211489484800643457487568896046642544492544023263042682880794844610560182044262402661335042236416819211265944312439406592
14101421602465863680000249604562807728128000223696248699635712000102632756516978688000278988455696670720004866582576612556800571354664277196800464517817860096002654316048384000106934275768320300963446784057699532800715653120516096016384710741169400928026624
Technical details
Node id
n-307dc0a9b33c92f9 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-1e7077c59c0f616e

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:            6,            2;
  2:           60,           36,            4;
  3:          840,          696,          144,            8;
  4:        15120,        15600,         4560,          480,          16;
  5:       332640,       404640,       149760,        22560,        1440,         32;
  6:      8648640,     11995200,      5302080,      1021440,       94080,       4032,        64;
  7:    259459200,    401143680,    204261120,     47376000,     5617920,     349440,     10752,     128;
  8:   8821612800,  14960090880,   8564209920,   2303562240,   330704640,   26611200,   1193472,   27648,   256;
  9: 335221286400, 616046860800, 389616998400, 118485158400, 19819537920, 1938746880, 112896000, 3824640, 69120, 512;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+2+2*x)*P(n-1,x) + 2*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*x)*P[n - 1, x] + 2*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*x)*v[i - 1] + 2*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.