Real-rooted atlas

A081578 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(9\,n\right)\,P_{n}(x) = \left(135 + 54\,x - 126\,n - 27\,n x + 36\,n^{2}\right)\,P_{n-1}(x) + \left(615 + 603\,x + 108\,x^{2} - 219\,n - 198\,n x - 27\,n x^{2}\right)\,P_{n-1}'(x) + \left(292 + 483\,x + 234\,x^{2} + 27\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1538
23739985
324538718927848
4166934712565783818569
511285292112898913635299724386360
6764532365832945971876236390910935729870829
751768518643712793501224642710410932762913863721878780192
835058611440198325225029245154871440026152111351129221133407656188528945
9237415251095591152196324452503511551782587258194178724229773442867545270919683892615592
101607779578233966951859157765243186421520357896051136621079427425093106596567168289651515591590499000032581
1110887880856127700451153876002852271770023521866805765143781041796575222061208647017144922510962375427501916517714790745188080
1273732759094523262846312504156621320566609249522380338272516945249758391369113669352840016799666852741182920023922657101316474671531441914962188841
1349931836565331614468219100070622393318145559664992202876162405188801113812311735086634375347419900031783552015894293809052372420646058093846435367554115943239225346460408
14338138493733241701887552779059669750451566826571015120994684136677200963445933511415307857382974399659834952929560225957858708134919184206248901280099521992851180965173781207478296993016977648925
Technical details
Node id
n-807761fed4e5ed8c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6004798a4a421655

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:        5,         3;
  2:       37,        39,         9;
  3:      245,       387,       189,        27;
  4:     1669,      3471,      2565,       783,        81;
  5:    11285,     29211,     28989,     13635,      2997,      243;
  6:    76453,    236583,    294597,    187623,     63909,    10935,      729;
  7:   517685,   1864371,   2793501,   2246427,   1041093,   276291,    38637,    2187;
  8:  3505861,  14401983,  25225029,  24515487,  14400261,  5211135,  1129221,  133407,   6561;
  9: 23741525, 109559115, 219632445, 250351155, 178258725, 81941787, 24229773, 4428675, 452709, 19683;
%F Formula
The row polynomials satisfy 9*n*P(n,x) = (36*n^2-126*n+135-(27*n-54)*x)*P(n-1,x) - (219*n-615+(198*n-603)*x+(27*n-108)*x^2)*d/dx P(n-1,x) + (292+483*x+234*x^2+27*x^3)*d^2/dx^2 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[((36*n^2 - 126*n + 135 - (27*n - 54)*x)*P[n - 1, x] - (219*n - 615 + (198*n - 603)*x + (27*n - 108)*x^2)*D[P[n - 1, x], x] + (292 + 483*x + 234*x^2 + 27*x^3)*D[P[n - 1, x], {x, 2}])/(9*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] = ((36*n^2 - 126*n + 135 - (27*n - 54)*x)*v[i - 1] - (219*n - 615 + (198*n - 603)*x + (27*n - 108)*x^2)*deriv(v[i - 1], x) + (292 + 483*x + 234*x^2 + 27*x^3)*deriv(deriv(v[i - 1], x), x))/(9*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

Candidate equalities ?

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