Real-rooted atlas

A304236 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-50\,n - 140\,n x + 96\,n x^{2} - 20\,n^{2} + 96\,n^{2} x + 24\,n^{3}\right)\,P_{n}(x) = \left(276 + 816\,x - 48\,x^{2} - 1152\,x^{3} - 230\,n - 1292\,n x - 1008\,n x^{2} + 1152\,n x^{3} - 260\,n^{2} + 144\,n^{2} x + 1152\,n^{2} x^{2} + 168\,n^{3} + 288\,n^{3} x\right)\,P_{n-1}(x) + \left(-13 - 388\,x - 1972\,x^{2} - 3648\,x^{3} - 2304\,x^{4} - 182\,n - 1324\,n x - 3072\,n x^{2} - 2304\,n x^{3} - 156\,n^{2} - 600\,n^{2} x - 576\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
210616
3191231
47312636235
5172348108628
6505148811882163397
712614356424886410729

All 15 rows

Technical details
Node id
n-bc3b39769daa4133 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a3fb6d37b7e5508a

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:    1;
  2:   10,     6;
  3:   19,    12;
  4:   73,   126,    36;
  5:  172,   348,   108;
  6:  505,  1488,  1188,   216;
  7: 1261,  4356,  4248,   864;
  8: 3358, 14298, 19980,  9936, 1296;
  9: 8443, 41196, 68796, 43200, 6480;
%F Formula
The row polynomials satisfy (24*n^3-20*n^2-50*n+(96*n^2-140*n)*x+96*n*x^2)*P(n,x) = (168*n^3-260*n^2-230*n+276+(288*n^3+144*n^2-1292*n+816)*x+(1152*n^2-1008*n-48)*x^2+(1152*n-1152)*x^3)*P(n-1,x) - (156*n^2+182*n+13+(600*n^2+1324*n+388)*x+(576*n^2+3072*n+1972)*x^2+(2304*n+3648)*x^3+2304*x^4)*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[((168*n^3 - 260*n^2 - 230*n + 276 + (288*n^3 + 144*n^2 - 1292*n + 816)*x + (1152*n^2 - 1008*n - 48)*x^2 + (1152*n - 1152)*x^3)*P[n - 1, x] - (156*n^2 + 182*n + 13 + (600*n^2 + 1324*n + 388)*x + (576*n^2 + 3072*n + 1972)*x^2 + (2304*n + 3648)*x^3 + 2304*x^4)*D[P[n - 1, x], x])/(24*n^3 - 20*n^2 - 50*n + (96*n^2 - 140*n)*x + 96*n*x^2)]];
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] = ((168*n^3 - 260*n^2 - 230*n + 276 + (288*n^3 + 144*n^2 - 1292*n + 816)*x + (1152*n^2 - 1008*n - 48)*x^2 + (1152*n - 1152)*x^3)*v[i - 1] - (156*n^2 + 182*n + 13 + (600*n^2 + 1324*n + 388)*x + (576*n^2 + 3072*n + 1972)*x^2 + (2304*n + 3648)*x^3 + 2304*x^4)*deriv(v[i - 1], x))/(24*n^3 - 20*n^2 - 50*n + (96*n^2 - 140*n)*x + 96*n*x^2)); 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.