Real-rooted atlas

A122542 → D+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(x^{3}\right)\,P_{n-6}(x) + \left(3\,x^{2} + 3\,x^{3}\right)\,P_{n-5}(x) + \left(3\,x + 3\,x^{2} + 3\,x^{3}\right)\,P_{n-4}(x) + \left(1 - 3\,x - 3\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(-3 - 3\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(3 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + x\)
\(P_{3}(x) = 6 + 6\,x + x^{2}\)
\(P_{4}(x) = 12 + 24\,x + 10\,x^{2} + x^{3}\)
\(P_{5}(x) = 20 + 70\,x + 56\,x^{2} + 14\,x^{3} + x^{4}\)
\(P_{6}(x) = 28 + 158\,x + 216\,x^{2} + 102\,x^{3} + 18\,x^{4} + x^{5}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
366113
4122410147
5207056141161
628158216102181523
7362946284941622211637
844478147217709482362614975

All 15 rows

Technical details
Node id
n-e8bf8c3e220bd2cf · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-6809751a249ad304

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:  2,   1;
   3:  6,   6,    1;
   4: 12,  24,   10,     1;
   5: 20,  70,   56,    14,     1;
   6: 28, 158,  216,   102,    18,    1;
   7: 36, 294,  628,   494,   162,   22,    1;
   8: 44, 478, 1472,  1770,   948,  236,   26,   1;
   9: 52, 710, 2940,  5038,  4052, 1622,  324,  30,  1;
  10: 60, 990, 5224, 12026, 13620, 8070, 2560, 426, 34, 1;
%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) = 1, P(2,x) = 2 + x, P(3,x) = 6 + 6*x + x^2, P(4,x) = 12 + 24*x + 10*x^2 + x^3, P(5,x) = 20 + 70*x + 56*x^2 + 14*x^3 + x^4, P(6,x) = 28 + 158*x + 216*x^2 + 102*x^3 + 18*x^4 + x^5.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[3, x_] := 6 + 6*x + x^2;
P[4, x_] := 12 + 24*x + 10*x^2 + x^3;
P[5, x_] := 20 + 70*x + 56*x^2 + 14*x^3 + x^4;
P[6, x_] := 28 + 158*x + 216*x^2 + 102*x^3 + 18*x^4 + x^5;
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] = 1; v[2] = 2 + x; v[3] = 6 + 6*x + x^2; v[4] = 12 + 24*x + 10*x^2 + x^3; v[5] = 20 + 70*x + 56*x^2 + 14*x^3 + x^4; v[6] = 28 + 158*x + 216*x^2 + 102*x^3 + 18*x^4 + x^5; 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.