Real-rooted atlas

A081577 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(3\,n\right)\,P_{n}(x) = \left(39 + 18\,x - 33\,n - 9\,n x + 9\,n^{2}\right)\,P_{n-1}(x) + \left(165 + 171\,x + 36\,x^{2} - 57\,n - 54\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x) + \left(76 + 129\,x + 66\,x^{2} + 9\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
23336978
320533317127736
4128927662142702816980
58085215912237411178267324366144
650737162072210375141480517599720729626872
73183651183809184662915621937716872218593426321875940992

All 15 rows

Technical details
Node id
n-8825354612a48a23 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-9f5d00811463c524

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:       33,       36,         9;
  3:      205,      333,       171,        27;
  4:     1289,     2766,      2142,       702,        81;
  5:     8085,    21591,     22374,     11178,      2673,      243;
  6:    50737,   162072,    210375,    141480,     51759,     9720,      729;
  7:   318365,  1183809,   1846629,   1562193,    771687,   221859,    34263,    2187;
  8:  1997721,  8475138,  15440004,  15744942,   9807966,  3812670,   901044,  118098,   6561;
  9: 12535525, 59749515, 124501320, 148622040, 111821310, 54892242, 17548488, 3516696, 400221, 19683;
%F Formula
The row polynomials satisfy 3*n*P(n,x) = (9*n^2-33*n+39-(9*n-18)*x)*P(n-1,x) - (57*n-165+(54*n-171)*x+(9*n-36)*x^2)*d/dx P(n-1,x) + (76+129*x+66*x^2+9*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[((9*n^2 - 33*n + 39 - (9*n - 18)*x)*P[n - 1, x] - (57*n - 165 + (54*n - 171)*x + (9*n - 36)*x^2)*D[P[n - 1, x], x] + (76 + 129*x + 66*x^2 + 9*x^3)*D[P[n - 1, x], {x, 2}])/(3*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] = ((9*n^2 - 33*n + 39 - (9*n - 18)*x)*v[i - 1] - (57*n - 165 + (54*n - 171)*x + (9*n - 36)*x^2)*deriv(v[i - 1], x) + (76 + 129*x + 66*x^2 + 9*x^3)*deriv(deriv(v[i - 1], x), x))/(3*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