Real-rooted atlas

A178820 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(15 + 9\,x + 5\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1201232
225030090640
325004500270054010240
4218755250047250189002835143360
5175000525000630000378000113400136081835008
6131250047250007087500567000025515006123606123622020096
7937500039375000708750007087500042525000153090003061800262440251658240
864453125309375000649687500779625000584718750280665000841995001443420010825652768240640
942968750023203125005568750000779625000070166250004209975000168399000043302600064953900433026029527900160
1027929687501675781250045246093750723937500007601343750054729675000273648375009382230000211100175028146690016888014307090161664
11177734375001173046875003519140625006334453125007601343750006385128750003831077250001641890250004925670750098513415001182160980644815083126736191488
1211108398437579980468750026393554687505278710937500712625976562568412093750004788846562500246283537500092356326562524628353750044331036750483611310024180565531267361914880
136835937500005332031250000191953125000004222968750000063344531250000684120937500005472967500000032837805000000147770122500004925670750000118216098000019344452400019344452400892820880307863255777280
14415039062500034863281250000135966796875000326320312500000538428515625000646114218750000581502796875000398744775000000209341006875000837364027500002512092082500054809281800008221392270007588977480032524189202990671627550720
Technical details
Node id
n-bbf993cd74f140c5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b1029f17d1b64336

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:        20,         12;
  2:       250,        300,         90;
  3:      2500,       4500,       2700,        540;
  4:     21875,      52500,      47250,      18900,       2835;
  5:    175000,     525000,     630000,     378000,     113400,      13608;
  6:   1312500,    4725000,    7087500,    5670000,    2551500,     612360,      61236;
  7:   9375000,   39375000,   70875000,   70875000,   42525000,   15309000,    3061800,    262440;
  8:  64453125,  309375000,  649687500,  779625000,  584718750,  280665000,   84199500,  14434200,  1082565;
  9: 429687500, 2320312500, 5568750000, 7796250000, 7016625000, 4209975000, 1683990000, 433026000, 64953900, 4330260;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+15+(3*n+9)*x)*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[((5*n + 15 + (3*n + 9)*x)*P[n - 1, 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] = ((5*n + 15 + (3*n + 9)*x)*v[i - 1])/(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

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