Real-rooted atlas

A128100 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-n + 7\,n x + 4\,n x^{2} + 6\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-24 - 56\,x - 38\,x^{2} - 8\,x^{3} - 2\,n + 35\,n x + 34\,n x^{2} + 8\,n x^{3} + 27\,n^{2} + 30\,n^{2} x + 8\,n^{2} x^{2} + 5\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-252 - 508\,x - 383\,x^{2} - 128\,x^{3} - 16\,x^{4} - 144\,n - 208\,n x - 100\,n x^{2} - 16\,n x^{3} - 18\,n^{2} - 17\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2415
3729
4189128
53722362
68465151165
7179162464391
8390417153221983
983310144418052373
101784245712632953015830
11379158103420956125614108
128042136099081300450539134281
1316989314462342389121806182782765
14358047201759343256456160798491199817
Technical details
Node id
n-9834c01ec55549ba · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-6f025d7ab02f4150

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:   4,    1;
  3:   7,    2;
  4:  18,    9,   1;
  5:  37,   22,   3;
  6:  84,   65,  15,  1;
  7: 179,  162,  46,  4;
  8: 390,  417, 153, 22, 1;
  9: 833, 1014, 441, 80, 5;
%F Formula
The row polynomials satisfy (n^3+6*n^2-n+(4*n^2+7*n)*x+4*n*x^2)*P(n,x) = (5*n^3+27*n^2-2*n-24+(2*n^3+30*n^2+35*n-56)*x+(8*n^2+34*n-38)*x^2+(8*n-8)*x^3)*P(n-1,x) - (18*n^2+144*n+252+(17*n^2+208*n+508)*x+(4*n^2+100*n+383)*x^2+(16*n+128)*x^3+16*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[((5*n^3 + 27*n^2 - 2*n - 24 + (2*n^3 + 30*n^2 + 35*n - 56)*x + (8*n^2 + 34*n - 38)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - (18*n^2 + 144*n + 252 + (17*n^2 + 208*n + 508)*x + (4*n^2 + 100*n + 383)*x^2 + (16*n + 128)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 + 6*n^2 - n + (4*n^2 + 7*n)*x + 4*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] = ((5*n^3 + 27*n^2 - 2*n - 24 + (2*n^3 + 30*n^2 + 35*n - 56)*x + (8*n^2 + 34*n - 38)*x^2 + (8*n - 8)*x^3)*v[i - 1] - (18*n^2 + 144*n + 252 + (17*n^2 + 208*n + 508)*x + (4*n^2 + 100*n + 383)*x^2 + (16*n + 128)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 + 6*n^2 - n + (4*n^2 + 7*n)*x + 4*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

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.