Real-rooted atlas

A081577 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6\,n - 6\,n x - 81\,n x^{2} + 36\,n x^{3} + 27\,n x^{4} + 18\,n^{2} + 9\,n^{2} x + 81\,n^{2} x^{2} + 27\,n^{2} x^{3} + 27\,n^{3} x\right)\,P_{n}(x) = \left(-72 - 420\,x - 540\,x^{2} + 333\,x^{3} + 378\,x^{4} + 81\,x^{5} + 24\,n + 354\,n x + 846\,n x^{2} + 117\,n x^{3} - 54\,n x^{4} + 162\,n^{2} + 387\,n^{2} x - 216\,n^{2} x^{2} - 135\,n^{2} x^{3} - 54\,n^{3} - 135\,n^{3} x\right)\,P_{n-1}(x) + \left(-8 - 36\,x + 114\,x^{2} + 855\,x^{3} + 1413\,x^{4} + 621\,x^{5} + 81\,x^{6} + 12\,n + 180\,n x + 801\,n x^{2} + 1296\,n x^{3} + 594\,n x^{4} + 81\,n x^{5} + 36\,n^{2} + 216\,n^{2} x + 351\,n^{2} x^{2} + 81\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
22436969
38825217127538
430813081656702814055
510205796111968748267324329676
63280234006141673440408249720729212809
710304886082955604844884108321759323426321871502174

All 15 rows

Technical details
Node id
n-9fef58ad95c2dd83 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a0b1595d36372e60

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:    24,      36,       9;
  3:    88,     252,     171,       27;
  4:   308,    1308,    1656,      702,       81;
  5:  1020,    5796,   11196,     8748,     2673,      243;
  6:  3280,   23400,   61416,    73440,    40824,     9720,     729;
  7: 10304,   88608,  295560,   484488,   410832,   175932,   34263,    2187;
  8: 31824,  320544, 1300320,  2729376,  3181356,  2067444,  717336,  118098,   6561;
  9: 96976, 1120176, 5359392, 13768704, 20636532, 18426204, 9649044, 2808108, 400221, 19683;
%F Formula
The row polynomials satisfy (18*n^2-6*n+(27*n^3+9*n^2-6*n)*x+(81*n^2-81*n)*x^2+(27*n^2+36*n)*x^3+27*n*x^4)*P(n,x) = -(54*n^3-162*n^2-24*n+72+(135*n^3-387*n^2-354*n+420)*x+(216*n^2-846*n+540)*x^2+(135*n^2-117*n-333)*x^3+(54*n-378)*x^4-81*x^5)*P(n-1,x) + (36*n^2+12*n-8+(216*n^2+180*n-36)*x+(351*n^2+801*n+114)*x^2+(81*n^2+1296*n+855)*x^3+(594*n+1413)*x^4+(81*n+621)*x^5+81*x^6)*d/dx 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[(-(54*n^3 - 162*n^2 - 24*n + 72 + (135*n^3 - 387*n^2 - 354*n + 420)*x + (216*n^2 - 846*n + 540)*x^2 + (135*n^2 - 117*n - 333)*x^3 + (54*n - 378)*x^4 - 81*x^5)*P[n - 1, x] + (36*n^2 + 12*n - 8 + (216*n^2 + 180*n - 36)*x + (351*n^2 + 801*n + 114)*x^2 + (81*n^2 + 1296*n + 855)*x^3 + (594*n + 1413)*x^4 + (81*n + 621)*x^5 + 81*x^6)*D[P[n - 1, x], x])/(18*n^2 - 6*n + (27*n^3 + 9*n^2 - 6*n)*x + (81*n^2 - 81*n)*x^2 + (27*n^2 + 36*n)*x^3 + 27*n*x^4)]];
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] = (-(54*n^3 - 162*n^2 - 24*n + 72 + (135*n^3 - 387*n^2 - 354*n + 420)*x + (216*n^2 - 846*n + 540)*x^2 + (135*n^2 - 117*n - 333)*x^3 + (54*n - 378)*x^4 - 81*x^5)*v[i - 1] + (36*n^2 + 12*n - 8 + (216*n^2 + 180*n - 36)*x + (351*n^2 + 801*n + 114)*x^2 + (81*n^2 + 1296*n + 855)*x^3 + (594*n + 1413)*x^4 + (81*n + 621)*x^5 + 81*x^6)*deriv(v[i - 1], x))/(18*n^2 - 6*n + (27*n^3 + 9*n^2 - 6*n)*x + (81*n^2 - 81*n)*x^2 + (27*n^2 + 36*n)*x^3 + 27*n*x^4)); 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.