Real-rooted atlas

A113413 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(24 + 45\,x + 27\,x^{2} - 36\,n - 36\,n x + 12\,n^{2} - 18\,n^{2} x - 27\,n^{2} x^{2} + 9\,n^{3} x\right)\,P_{n}(x) = \left(48 + 144\,x + 54\,x^{2} - 168\,n - 675\,n x - 729\,n x^{2} - 243\,n x^{3} + 96\,n^{2} + 315\,n^{2} x + 189\,n^{2} x^{2} - 12\,n^{3} - 27\,n^{3} x\right)\,P_{n-1}(x) + \left(8 + 48\,x + 81\,x^{2} + 27\,x^{3} - 24\,n - 168\,n x - 387\,n x^{2} - 324\,n x^{3} - 81\,n x^{4} + 8\,n^{2} + 48\,n^{2} x + 81\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2181836
36916281312
42288649723242388
5699367266424860121517088
62046137703466840500218704374117228
75805476821544672511002150559185415309781272
816104156240620136129924015454801041012367416524885098116
94392349161623123885947992911979085118044715172141717617714732737008
101182181499634816026424925320469621805645959243232616203128565314410590490207575580
1131480544637302759354197689240218819070323326080318580290206435304841448251948617019486171302801672
1283098813025232901824843633190209444049201659924252200964304816736848569388791003377602807015021263772928108181684
1321775473738895228665883812954313803836409885783078094811279242548115362850328307308565410508648013211623262487143882072619950107373088
145670774105854346890363988446154706814837538690345324194465778559634470308148392621371446023936855879017373869172505719255687050035866961566307801366092
Technical details
Node id
n-591fc3f741366e13 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f760ab6df97a669b

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     3;
  2:    18,     18;
  3:    69,    162,      81;
  4:   228,    864,     972,     324;
  5:   699,   3672,    6642,    4860,    1215;
  6:  2046,  13770,   34668,   40500,   21870,    4374;
  7:  5805,  47682,  154467,  251100,  215055,   91854,   15309;
  8: 16104, 156240,  620136, 1299240, 1545480, 1041012,  367416,   52488;
  9: 43923, 491616, 2312388, 5947992, 9119790, 8511804, 4715172, 1417176, 177147;
%F Formula
The row polynomials satisfy (12*n^2-36*n+24+(9*n^3-18*n^2-36*n+45)*x-(27*n^2-27)*x^2)*P(n,x) = -(12*n^3-96*n^2+168*n-48+(27*n^3-315*n^2+675*n-144)*x-(189*n^2-729*n+54)*x^2+243*n*x^3)*P(n-1,x) + (8*n^2-24*n+8+(48*n^2-168*n+48)*x+(81*n^2-387*n+81)*x^2+(27*n^2-324*n+27)*x^3-81*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n^3 - 96*n^2 + 168*n - 48 + (27*n^3 - 315*n^2 + 675*n - 144)*x - (189*n^2 - 729*n + 54)*x^2 + 243*n*x^3)*P[n - 1, x] + (8*n^2 - 24*n + 8 + (48*n^2 - 168*n + 48)*x + (81*n^2 - 387*n + 81)*x^2 + (27*n^2 - 324*n + 27)*x^3 - 81*n*x^4)*D[P[n - 1, x], x])/(12*n^2 - 36*n + 24 + (9*n^3 - 18*n^2 - 36*n + 45)*x - (27*n^2 - 27)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n^3 - 96*n^2 + 168*n - 48 + (27*n^3 - 315*n^2 + 675*n - 144)*x - (189*n^2 - 729*n + 54)*x^2 + 243*n*x^3)*v[i - 1] + (8*n^2 - 24*n + 8 + (48*n^2 - 168*n + 48)*x + (81*n^2 - 387*n + 81)*x^2 + (27*n^2 - 324*n + 27)*x^3 - 81*n*x^4)*deriv(v[i - 1], x))/(12*n^2 - 36*n + 24 + (9*n^3 - 18*n^2 - 36*n + 45)*x - (27*n^2 - 27)*x^2)); 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

References

  1. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.