Real-rooted atlas

A201637 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-7 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(10 - 79\,x + 275\,x^{2} - 128\,x^{3} + 12\,x^{4} - 4\,n + 50\,n x - 246\,n x^{2} + 78\,n x^{3} - 4\,n x^{4} - 8\,n^{2} x + 76\,n^{2} x^{2} - 12\,n^{2} x^{3} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(5\,x - 42\,x^{2} + 56\,x^{3} - 21\,x^{4} + 2\,x^{5} - 2\,n x + 26\,n x^{2} - 30\,n x^{3} + 6\,n x^{4} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-19 + 83\,x - 31\,x^{2} + 2\,x^{3} + 6\,n - 50\,n x + 10\,n x^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-7\,x + 9\,x^{2} - 2\,x^{3} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
111
211
3325
4920635
52313813024315
65370816609241203465
711530181465219232730872045045
824111460101970264680228060642245040675675

All 15 rows

Technical details
Node id
n-17f334db6f3feea7 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7af5d5c3f437902e

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:    1;
   3:    3,      2;
   4:    9,     20,       6;
   5:   23,    138,     130,       24;
   6:   53,    708,    1660,      924,      120;
   7:  115,   3018,   14652,    19232,     7308,      720;
   8:  241,  11460,  101970,   264680,   228060,    64224,     5040;
   9:  495,  40394,  607350,  2771880,  4570540,  2832120,   623376,   40320;
  10: 1005, 135524, 3254760, 24124500, 67965820, 78597840, 37163736, 6636960, 362880;
%F Formula
The row polynomials satisfy (2*n-7+2*x)*P(n,x) = -(4*n-10+(8*n^2-50*n+79)*x+(8*n^3-76*n^2+246*n-275)*x^2+(12*n^2-78*n+128)*x^3+(4*n-12)*x^4)*P(n-2,x) - ((2*n-5)*x+(4*n^2-26*n+42)*x^2-(4*n^2-30*n+56)*x^3-(6*n-21)*x^4-2*x^5)*d/dx P(n-2,x) + (6*n-19+(8*n^2-50*n+83)*x+(10*n-31)*x^2+2*x^3)*P(n-1,x) + ((2*n-7)*x-(2*n-9)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 10 + (8*n^2 - 50*n + 79)*x + (8*n^3 - 76*n^2 + 246*n - 275)*x^2 + (12*n^2 - 78*n + 128)*x^3 + (4*n - 12)*x^4)*P[n - 2, x] - ((2*n - 5)*x + (4*n^2 - 26*n + 42)*x^2 - (4*n^2 - 30*n + 56)*x^3 - (6*n - 21)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (6*n - 19 + (8*n^2 - 50*n + 83)*x + (10*n - 31)*x^2 + 2*x^3)*P[n - 1, x] + ((2*n - 7)*x - (2*n - 9)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(2*n - 7 + 2*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] = 1; for(n = 3, N, my(i = n); v[i] = (-(4*n - 10 + (8*n^2 - 50*n + 79)*x + (8*n^3 - 76*n^2 + 246*n - 275)*x^2 + (12*n^2 - 78*n + 128)*x^3 + (4*n - 12)*x^4)*v[i - 2] - ((2*n - 5)*x + (4*n^2 - 26*n + 42)*x^2 - (4*n^2 - 30*n + 56)*x^3 - (6*n - 21)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (6*n - 19 + (8*n^2 - 50*n + 83)*x + (10*n - 31)*x^2 + 2*x^3)*v[i - 1] + ((2*n - 7)*x - (2*n - 9)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(2*n - 7 + 2*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

Same fingerprint ?

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.