Real-rooted atlas

A122542 → D → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(x^{3}\right)\,P_{n-6}(x) + \left(3\,x^{2} + 3\,x^{3}\right)\,P_{n-5}(x) + \left(3\,x + 3\,x^{2} + 3\,x^{3}\right)\,P_{n-4}(x) + \left(1 - 3\,x - 3\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(-3 - 3\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(3 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 4 + 2\,x\)
\(P_{4}(x) = 10 + 14\,x + 3\,x^{2}\)
\(P_{5}(x) = 18 + 52\,x + 30\,x^{2} + 4\,x^{3}\)
\(P_{6}(x) = 26 + 132\,x + 150\,x^{2} + 52\,x^{3} + 5\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3426
41014327
51852304104
626132150525365
7342604983288061206
8424361254135261011473815

All 15 rows

Technical details
Node id
n-7b9ff4636de07cf2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-30a879bddb7f3b46

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 -2)
  • 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:  0;
   2:  1;
   3:  4,   2;
   4: 10,  14,    3;
   5: 18,  52,   30,     4;
   6: 26, 132,  150,    52,     5;
   7: 34, 260,  498,   328,    80,    6;
   8: 42, 436, 1254,  1352,   610,  114,    7;
   9: 50, 660, 2610,  4168,  3010, 1020,  154,   8;
  10: 58, 932, 4758, 10440, 11010, 5868, 1582, 200, 9;
%F Formula
The row polynomials satisfy P(n,x) = x^3*P(n-6,x) + (3*x^2+3*x^3)*P(n-5,x) + (3*x+3*x^2+3*x^3)*P(n-4,x) + (1-3*x-3*x^2+x^3)*P(n-3,x) - (3+3*x+3*x^2)*P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 7, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 4 + 2*x, P(4,x) = 10 + 14*x + 3*x^2, P(5,x) = 18 + 52*x + 30*x^2 + 4*x^3, P(6,x) = 26 + 132*x + 150*x^2 + 52*x^3 + 5*x^4.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 4 + 2*x;
P[4, x_] := 10 + 14*x + 3*x^2;
P[5, x_] := 18 + 52*x + 30*x^2 + 4*x^3;
P[6, x_] := 26 + 132*x + 150*x^2 + 52*x^3 + 5*x^4;
P[n_, x_] := P[n, x] = Expand[x^3*P[n - 6, x] + (3*x^2 + 3*x^3)*P[n - 5, x] + (3*x + 3*x^2 + 3*x^3)*P[n - 4, x] + (1 - 3*x - 3*x^2 + x^3)*P[n - 3, x] - (3 + 3*x + 3*x^2)*P[n - 2, x] + (3 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 4 + 2*x; v[4] = 10 + 14*x + 3*x^2; v[5] = 18 + 52*x + 30*x^2 + 4*x^3; v[6] = 26 + 132*x + 150*x^2 + 52*x^3 + 5*x^4; for(n = 7, N, my(i = n); v[i] = x^3*v[i - 6] + (3*x^2 + 3*x^3)*v[i - 5] + (3*x + 3*x^2 + 3*x^3)*v[i - 4] + (1 - 3*x - 3*x^2 + x^3)*v[i - 3] - (3 + 3*x + 3*x^2)*v[i - 2] + (3 + 3*x)*v[i - 1]); 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. [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.