Real-rooted atlas

A046739 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2\,n x^{2} - n^{2} x^{2}\right)\,P_{n-4}(x) + \left(-4\,n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-2\,x^{2} + 2\,x^{3} - n x^{2} + n x^{3}\right)\,P_{n-3}'(x) + \left(4\,x + 2\,n x - 2\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} + 2\,x^{3} - n x^{2} + n x^{3}\right)\,P_{n-2}'(x) + \left(1 + 2\,x + 2\,n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 14\,x + 3\,x^{2}\)
\(P_{2}(x) = 1 + 42\,x + 63\,x^{2} + 4\,x^{3}\)
\(P_{3}(x) = 1 + 102\,x + 483\,x^{2} + 204\,x^{3} + 5\,x^{4}\)
\(P_{4}(x) = 1 + 226\,x + 2439\,x^{2} + 3252\,x^{3} + 565\,x^{4} + 6\,x^{5}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1114318
2142634110
311024832045795
412262439325256566489
514781008330524168051434759332
61986372632153322691657452634518600732
712006128595128042830350451920642300055802496674805
81405042464768187722747008532964102119328511132392182251080765135

All 15 rows

Technical details
Node id
n-40ed3d5c7bd8f231 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9ca2e07920e8f820

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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,    14,       3;
   2: 1,    42,      63,         4;
   3: 1,   102,     483,       204,          5;
   4: 1,   226,    2439,      3252,        565,          6;
   5: 1,   478,   10083,     30524,      16805,       1434,          7;
   6: 1,   986,   37263,    215332,     269165,      74526,       3451,          8;
   7: 1,  2006,  128595,   1280428,    3035045,    1920642,     300055,       8024,         9;
   8: 1,  4050,  424647,   6818772,   27470085,   32964102,   11932851,    1132392,     18225,       10;
   9: 1,  8142, 1361715,  33690716,  214620565,  435631818,  300468791,   67381432,   4085145,    40710,    11;
  10: 1, 16330, 4280175, 157897796, 1512291565, 4812172878, 5614201691, 2419666504, 355270041, 14267250, 89815, 12;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n)*x^2*P(n-4,x) - (2*n^2+4*n)*x^2*P(n-3,x) - ((n+2)*x^2-(n+2)*x^3)*d/dx P(n-3,x) + ((2*n+4)*x-(n^2+2*n)*x^2)*P(n-2,x) - ((n+2)*x^2-(n+2)*x^3)*d/dx P(n-2,x) + (1+(2*n+2)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1 + 14*x + 3*x^2, P(2,x) = 1 + 42*x + 63*x^2 + 4*x^3, P(3,x) = 1 + 102*x + 483*x^2 + 204*x^3 + 5*x^4, P(4,x) = 1 + 226*x + 2439*x^2 + 3252*x^3 + 565*x^4 + 6*x^5.
%t Mathematica
P[1, x_] := 1 + 14*x + 3*x^2;
P[2, x_] := 1 + 42*x + 63*x^2 + 4*x^3;
P[3, x_] := 1 + 102*x + 483*x^2 + 204*x^3 + 5*x^4;
P[4, x_] := 1 + 226*x + 2439*x^2 + 3252*x^3 + 565*x^4 + 6*x^5;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n)*x^2*P[n - 4, x] - (2*n^2 + 4*n)*x^2*P[n - 3, x] - ((n + 2)*x^2 - (n + 2)*x^3)*D[P[n - 3, x], x] + ((2*n + 4)*x - (n^2 + 2*n)*x^2)*P[n - 2, x] - ((n + 2)*x^2 - (n + 2)*x^3)*D[P[n - 2, x], x] + (1 + (2*n + 2)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 14*x + 3*x^2; v[2] = 1 + 42*x + 63*x^2 + 4*x^3; v[3] = 1 + 102*x + 483*x^2 + 204*x^3 + 5*x^4; v[4] = 1 + 226*x + 2439*x^2 + 3252*x^3 + 565*x^4 + 6*x^5; for(n = 5, N, my(i = n); v[i] = -(n^2 + 2*n)*x^2*v[i - 4] - (2*n^2 + 4*n)*x^2*v[i - 3] - ((n + 2)*x^2 - (n + 2)*x^3)*deriv(v[i - 3], x) + ((2*n + 4)*x - (n^2 + 2*n)*x^2)*v[i - 2] - ((n + 2)*x^2 - (n + 2)*x^3)*deriv(v[i - 2], x) + (1 + (2*n + 2)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.