Real-rooted atlas

A056242 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6 + 6\,x + 14\,x^{2} - 4\,x^{3} + 15\,n - 22\,n x^{2} + 4\,n x^{3} - 12\,n^{2} - 10\,n^{2} x + 8\,n^{2} x^{2} + 3\,n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(16\,x + 48\,x^{2} + 42\,x^{3} - 20\,x^{4} + 12\,n + 13\,n x - 42\,n x^{2} - 62\,n x^{3} + 12\,n x^{4} - 18\,n^{2} - 46\,n^{2} x - 14\,n^{2} x^{2} + 24\,n^{2} x^{3} + 6\,n^{3} + 17\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(8\,x^{2} + 20\,x^{3} + 4\,x^{4} - 8\,x^{5} + 6\,n x + 6\,n x^{2} - 16\,n x^{3} - 16\,n x^{4} - 6\,n^{2} x - 14\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
397117
4243412171
56012683181288
61444044011652511140
7336118415879972903314428
8768326455364740212846942116948

All 15 rows

Technical details
Node id
n-d88b91eb9ff3e6dc · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-18761d0c01c9a753

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    3,     1;
   3:    9,     7,     1;
   4:   24,    34,    12,     1;
   5:   60,   126,    83,    18,     1;
   6:  144,   404,   401,   165,    25,     1;
   7:  336,  1184,  1587,   997,   290,    33,    1;
   8:  768,  3264,  5536,  4740,  2128,   469,   42,    1;
   9: 1728,  8608, 17704, 19308, 11909,  4088,  714,   52,  1;
  10: 3840, 21952, 53136, 70616, 55807, 26505, 7266, 1038, 63, 1;
%F Formula
The row polynomials satisfy (3*n^3-12*n^2+15*n-6+(4*n^3-10*n^2+6)*x+(8*n^2-22*n+14)*x^2+(4*n-4)*x^3)*P(n,x) = (6*n^3-18*n^2+12*n+(17*n^3-46*n^2+13*n+16)*x+(12*n^3-14*n^2-42*n+48)*x^2+(24*n^2-62*n+42)*x^3+(12*n-20)*x^4)*P(n-1,x) - ((6*n^2-6*n)*x+(14*n^2-6*n-8)*x^2+(8*n^2+16*n-20)*x^3+(16*n-4)*x^4+8*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((6*n^3 - 18*n^2 + 12*n + (17*n^3 - 46*n^2 + 13*n + 16)*x + (12*n^3 - 14*n^2 - 42*n + 48)*x^2 + (24*n^2 - 62*n + 42)*x^3 + (12*n - 20)*x^4)*P[n - 1, x] - ((6*n^2 - 6*n)*x + (14*n^2 - 6*n - 8)*x^2 + (8*n^2 + 16*n - 20)*x^3 + (16*n - 4)*x^4 + 8*x^5)*D[P[n - 1, x], x])/(3*n^3 - 12*n^2 + 15*n - 6 + (4*n^3 - 10*n^2 + 6)*x + (8*n^2 - 22*n + 14)*x^2 + (4*n - 4)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((6*n^3 - 18*n^2 + 12*n + (17*n^3 - 46*n^2 + 13*n + 16)*x + (12*n^3 - 14*n^2 - 42*n + 48)*x^2 + (24*n^2 - 62*n + 42)*x^3 + (12*n - 20)*x^4)*v[i - 1] - ((6*n^2 - 6*n)*x + (14*n^2 - 6*n - 8)*x^2 + (8*n^2 + 16*n - 20)*x^3 + (16*n - 4)*x^4 + 8*x^5)*deriv(v[i - 1], x))/(3*n^3 - 12*n^2 + 15*n - 6 + (4*n^3 - 10*n^2 + 6)*x + (8*n^2 - 22*n + 14)*x^2 + (4*n - 4)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

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