Real-rooted atlas

A100861 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 - 6\,x + 4\,x^{2} - n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(-2 - 2\,x + 4\,x^{2} - 8\,x^{3} + n - 2\,n x - 8\,n x^{2} + 8\,n x^{3} - n^{2} + 8\,n^{2} x^{2} + n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-1 - 6\,x - 16\,x^{2} - 24\,x^{3} - 16\,x^{4} - n - 8\,n x - 20\,n x^{2} - 16\,n x^{3} - n^{2} - 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2426
310616
434481294
510620060366
637610207201202236
713244452420084010816
85020210562856013440168069756
9193249532816027210080015120390844
1078256451800947520856800302400302402667016
11323896212564052430405876640277200033264016673856
121393624103297922981880041690880282744007983360665280120156136
13613708050527776165799920272638080229188960864864008648640819426856
1427898144253639568940011072180900720018969350401029188160242161920172972806216138384
Technical details
Node id
n-913cf18637a4cc4e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-c9830293e85b16de

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,     2;
  3:    10,     6;
  4:    34,    48,     12;
  5:   106,   200,     60;
  6:   376,  1020,    720,    120;
  7:  1324,  4452,   4200,    840;
  8:  5020, 21056,  28560,  13440,  1680;
  9: 19324, 95328, 160272, 100800, 15120;
%F Formula
The row polynomials satisfy (n^2-n-1+(4*n-6)*x+4*x^2)*P(n,x) = (n^3-n^2+n-2+(2*n^3-2*n-2)*x+(8*n^2-8*n+4)*x^2+(8*n-8)*x^3)*P(n-1,x) - (n^2+n+1+(4*n^2+8*n+6)*x+(4*n^2+20*n+16)*x^2+(16*n+24)*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[((n^3 - n^2 + n - 2 + (2*n^3 - 2*n - 2)*x + (8*n^2 - 8*n + 4)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - (n^2 + n + 1 + (4*n^2 + 8*n + 6)*x + (4*n^2 + 20*n + 16)*x^2 + (16*n + 24)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^2 - n - 1 + (4*n - 6)*x + 4*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] = ((n^3 - n^2 + n - 2 + (2*n^3 - 2*n - 2)*x + (8*n^2 - 8*n + 4)*x^2 + (8*n - 8)*x^3)*v[i - 1] - (n^2 + n + 1 + (4*n^2 + 8*n + 6)*x + (4*n^2 + 20*n + 16)*x^2 + (16*n + 24)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^2 - n - 1 + (4*n - 6)*x + 4*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.