Real-rooted atlas

A098158 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4 - 60\,x + 52\,x^{2} - 8\,x^{3} - 8\,n + 74\,n x - 42\,n x^{2} + 4\,n x^{3} + 5\,n^{2} - 30\,n^{2} x + 8\,n^{2} x^{2} - n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(-8 + 60\,x - 156\,x^{2} + 140\,x^{3} - 24\,x^{4} + 14\,n - 92\,n x + 210\,n x^{2} - 118\,n x^{3} + 12\,n x^{4} - 7\,n^{2} + 45\,n^{2} x - 90\,n^{2} x^{2} + 24\,n^{2} x^{3} + n^{3} - 7\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x + 32\,x^{2} - 64\,x^{3} + 44\,x^{4} - 8\,x^{5} + 6\,n x - 38\,n x^{2} + 48\,n x^{3} - 16\,n x^{4} - 2\,n^{2} x + 10\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2112
31315
4066113
5021910132
600154515176
70036190211176
800028182161281400
90004141448266361896
100000455139664144511984
1100005271151518906155514352
1200000661166386134328806619472
13000006463402688115874122178120480
1400000091230111869184479581165191144032
15000000772991003103136036150152184105194208
Technical details
Node id
n-a5498e309d62eadd · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-771adcd1557a8401

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 -1)
  • 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, 1;
   3: 1, 3,  1;
   4: 0, 6,  6,  1;
   5: 0, 2, 19, 10,   1;
   6: 0, 0, 15, 45,  15,   1;
   7: 0, 0,  3, 61,  90,  21,   1;
   8: 0, 0,  0, 28, 182, 161,  28,   1;
   9: 0, 0,  0,  4, 141, 448, 266,  36,  1;
  10: 0, 0,  0,  0,  45, 513, 966, 414, 45, 1;
%F Formula
The row polynomials satisfy -(n^3-5*n^2+8*n-4-(4*n^3-30*n^2+74*n-60)*x-(8*n^2-42*n+52)*x^2-(4*n-8)*x^3)*P(n,x) = (n^3-7*n^2+14*n-8-(7*n^3-45*n^2+92*n-60)*x+(12*n^3-90*n^2+210*n-156)*x^2+(24*n^2-118*n+140)*x^3+(12*n-24)*x^4)*P(n-1,x) - ((2*n^2-6*n+4)*x-(10*n^2-38*n+32)*x^2+(8*n^2-48*n+64)*x^3+(16*n-44)*x^4+8*x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 7*n^2 + 14*n - 8 - (7*n^3 - 45*n^2 + 92*n - 60)*x + (12*n^3 - 90*n^2 + 210*n - 156)*x^2 + (24*n^2 - 118*n + 140)*x^3 + (12*n - 24)*x^4)*P[n - 1, x] - ((2*n^2 - 6*n + 4)*x - (10*n^2 - 38*n + 32)*x^2 + (8*n^2 - 48*n + 64)*x^3 + (16*n - 44)*x^4 + 8*x^5)*D[P[n - 1, x], x])/(-n^3 + 5*n^2 - 8*n + 4 + (4*n^3 - 30*n^2 + 74*n - 60)*x + (8*n^2 - 42*n + 52)*x^2 + (4*n - 8)*x^3)]];
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 + x; for(n = 3, N, my(i = n); v[i] = ((n^3 - 7*n^2 + 14*n - 8 - (7*n^3 - 45*n^2 + 92*n - 60)*x + (12*n^3 - 90*n^2 + 210*n - 156)*x^2 + (24*n^2 - 118*n + 140)*x^3 + (12*n - 24)*x^4)*v[i - 1] - ((2*n^2 - 6*n + 4)*x - (10*n^2 - 38*n + 32)*x^2 + (8*n^2 - 48*n + 64)*x^3 + (16*n - 44)*x^4 + 8*x^5)*deriv(v[i - 1], x))/(-n^3 + 5*n^2 - 8*n + 4 + (4*n^3 - 30*n^2 + 74*n - 60)*x + (8*n^2 - 42*n + 52)*x^2 + (4*n - 8)*x^3)); 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

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.