Real-rooted atlas

A166979 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 3\,x + 2\,x^{2} - 9\,n x - 18\,n x^{2} - 9\,n x^{3}\right)\,P_{n-1}(x) + \left(9\,x + 27\,x^{2} + 27\,x^{3} + 9\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 13\,x + 13\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
11131327
21114255143513
31111540204757185211745
4111116585041111319130427563299619
511111178336312301964302108119153564737258656875
61111111812060957461166318637320986374968444589459290396285786225
7111111119181380046929741371233024925132631551112618039404112940633520374523510666827873
811111111202868442614191658363276154883786711292727191312536629640483662314795309954476434912490375441558049059
911111111121477226312654053003841141617449208222372885748127153676485892484924628953728280990647039691160664123512877767233819943009142939
1011111111112282895221547587768339286114262937819325812011433260773221533613012719425473063816635719902372336764499141525172790285993431670203643086083981972835054141185
1111111111111231487231431528819418526485615311327380047667983827592802264230831703600256629212185284313228454122257944569386358313417088459196367309883350378295350663815054011055760597890150948194436139393
12111111111111242727961942115404385125413900947304751641154756501218271702887350553146383589617089245553539215091071506870687374825096675979425965220417723596747503527682781426977161059355763365761075848181335797769143242502851574835730311747
131111111111111255075349912240115101112612246513464833081598797698823389895648906527435011790942192322086795245686262030619711723644269435823572212938967174540308419547299156111319249415671612269690872999187858013717499424269191253273472590084245123074955978249433169823702674226731563
141111111111111126952538705547828092383222350625245247231979985077369815511260216067471174483684409599560577025517551067921526304525543938186050610271014140852123763387486771006615982235167750192160630417153402105994691703593774846709124044603673124629965422462343675249065909895183260399447413311190516921910714920124947851304369
151111111111111112717970457627630937465747660037369293256765890667623196435422141245279875279202636411653726664078162889315245981034174806500665037926237383730347373223121384349946274079705812311551022045083836278174661301326978251598174970078071173308311099697081935961369613373573291677108990241698831981908882013931604702689139729171363047274358839412713412067088926550964033
Technical details
Node id
n-f83cea3351353f16 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-732c894f3f2377d3

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1,          13,           13;
   2: 1,         114,          255,           143;
   3: 1,        1115,         4020,          4757,           1852;
   4: 1,       11116,        58504,        111131,          91304,           27563;
   5: 1,      111117,       833631,       2301964,        3021081,         1915356,          473725;
   6: 1,     1111118,     12060957,      46116631,       86373209,        86374968,        44458945,         9290396;
   7: 1,    11111119,    181380046,     929741371,     2330249251,      3263155111,      2618039404,      1129406335,       203745235;
   8: 1,   111111120,   2868442614,   19165836327,    61548837867,    112927271913,    125366296404,     83662314795,     30995447643,     4912490375;
   9: 1,  1111111121,  47722631265,  405300384114,  1617449208222,   3728857481271,   5367648589248,   4924628953728,   2809906470396,   911606641235,   128777672338;
  10: 1, 11111111122, 828952215475, 8776833928611, 42629378193258, 120114332607732, 215336130127194, 254730638166357, 199023723367644, 99141525172790, 28599343167020, 3643086083981;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (1-(9*n-3)*x-(18*n-2)*x^2-9*n*x^3)*P(n-1,x) + (9*x+27*x^2+27*x^3+9*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 13*x + 13*x^2.
%t Mathematica
P[1, x_] := 1 + 13*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 - (9*n - 3)*x - (18*n - 2)*x^2 - 9*n*x^3)*P[n - 1, x] + (9*x + 27*x^2 + 27*x^3 + 9*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 13*x + 13*x^2; for(n = 2, N, my(i = n); v[i] = ((1 - (9*n - 3)*x - (18*n - 2)*x^2 - 9*n*x^3)*v[i - 1] + (9*x + 27*x^2 + 27*x^3 + 9*x^4)*deriv(v[i - 1], x))/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

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