Real-rooted atlas

A116412 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-39 - 36\,x + 16\,x^{2} + 10\,x^{3} + x^{4} + 23\,n + 38\,n x + 3\,n x^{2} + 7\,n x^{3} + n x^{4} + 15\,n^{2} + n^{2} x + 7\,n^{2} x^{2} + n^{2} x^{3} - 3\,n^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(-214 - 218\,x + 80\,x^{2} + 128\,x^{3} + 30\,x^{4} + 2\,x^{5} + 170\,n + 260\,n x + 84\,n x^{2} - 8\,n x^{3} - 2\,n x^{4} + 48\,n^{2} - 4\,n^{2} x - 28\,n^{2} x^{2} - 4\,n^{2} x^{3} - 24\,n^{3} - 12\,n^{3} x\right)\,P_{n-1}(x) + \left(-39 + 22\,x + 245\,x^{2} + 294\,x^{3} + 129\,x^{4} + 20\,x^{5} + x^{6} + 78\,n + 242\,n x + 263\,n x^{2} + 117\,n x^{3} + 19\,n x^{4} + n x^{5} + 39\,n^{2} + 69\,n^{2} x + 33\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1516
22110132
38566151167
4332364135201852
5126618079492282514276
6474183705745195234530121184
7175073691331455139783485486351103860

All 15 rows

Technical details
Node id
n-6207009db611b044 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f7624da2329372ca

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      5,      1;
  2:     21,     10,      1;
  3:     85,     66,     15,      1;
  4:    332,    364,    135,     20,      1;
  5:   1266,   1807,    949,    228,     25,     1;
  6:   4741,   8370,   5745,   1952,    345,    30,    1;
  7:  17507,  36913,  31455,  13978,   3485,   486,   35,   1;
  8:  63927, 156940, 160243,  88536,  28850,  5660,  651,  40,  1;
  9: 231301, 648588, 773253, 512778, 207804, 53217, 8589, 840, 45, 1;
%F Formula
The row polynomials satisfy -(3*n^3-15*n^2-23*n+39-(3*n^3+n^2+38*n-36)*x-(7*n^2+3*n+16)*x^2-(n^2+7*n+10)*x^3-(n+1)*x^4)*P(n,x) = -(24*n^3-48*n^2-170*n+214+(12*n^3+4*n^2-260*n+218)*x+(28*n^2-84*n-80)*x^2+(4*n^2+8*n-128)*x^3+(2*n-30)*x^4-2*x^5)*P(n-1,x) + (39*n^2+78*n-39+(69*n^2+242*n+22)*x+(33*n^2+263*n+245)*x^2+(3*n^2+117*n+294)*x^3+(19*n+129)*x^4+(n+20)*x^5+x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n^3 - 48*n^2 - 170*n + 214 + (12*n^3 + 4*n^2 - 260*n + 218)*x + (28*n^2 - 84*n - 80)*x^2 + (4*n^2 + 8*n - 128)*x^3 + (2*n - 30)*x^4 - 2*x^5)*P[n - 1, x] + (39*n^2 + 78*n - 39 + (69*n^2 + 242*n + 22)*x + (33*n^2 + 263*n + 245)*x^2 + (3*n^2 + 117*n + 294)*x^3 + (19*n + 129)*x^4 + (n + 20)*x^5 + x^6)*D[P[n - 1, x], x])/(-3*n^3 + 15*n^2 + 23*n - 39 + (3*n^3 + n^2 + 38*n - 36)*x + (7*n^2 + 3*n + 16)*x^2 + (n^2 + 7*n + 10)*x^3 + (n + 1)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(24*n^3 - 48*n^2 - 170*n + 214 + (12*n^3 + 4*n^2 - 260*n + 218)*x + (28*n^2 - 84*n - 80)*x^2 + (4*n^2 + 8*n - 128)*x^3 + (2*n - 30)*x^4 - 2*x^5)*v[i - 1] + (39*n^2 + 78*n - 39 + (69*n^2 + 242*n + 22)*x + (33*n^2 + 263*n + 245)*x^2 + (3*n^2 + 117*n + 294)*x^3 + (19*n + 129)*x^4 + (n + 20)*x^5 + x^6)*deriv(v[i - 1], x))/(-3*n^3 + 15*n^2 + 23*n - 39 + (3*n^3 + n^2 + 38*n - 36)*x + (7*n^2 + 3*n + 16)*x^2 + (n^2 + 7*n + 10)*x^3 + (n + 1)*x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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.