Real-rooted atlas

A081577 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n + 6\,n x + 5\,n x^{2} + 10\,n x^{3} + n x^{4} + 6\,n^{2} + 9\,n^{2} x + 11\,n^{2} x^{2} + n^{2} x^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(-24 + 12\,x + 112\,x^{2} + 93\,x^{3} + 18\,x^{4} + x^{5} + 72\,n + 114\,n x + 6\,n x^{2} - 39\,n x^{3} - 4\,n x^{4} + 6\,n^{2} - 45\,n^{2} x - 52\,n^{2} x^{2} - 5\,n^{2} x^{3} - 18\,n^{3} - 15\,n^{3} x\right)\,P_{n-1}(x) + \left(72 + 348\,x + 630\,x^{2} + 527\,x^{3} + 197\,x^{4} + 25\,x^{5} + x^{6} + 108\,n + 348\,n x + 393\,n x^{2} + 176\,n x^{3} + 24\,n x^{4} + n x^{5} + 36\,n^{2} + 72\,n^{2} x + 39\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2128121
3404413198
4132204100181455
54208445961802312064
613123256303213282842819241
7403211936138808088251241233140894

All 15 rows

Technical details
Node id
n-28b53d702f06d6e0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-54a229d497821396

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:     3,      1;
  2:    12,      8,      1;
  3:    40,     44,     13,      1;
  4:   132,    204,    100,     18,      1;
  5:   420,    844,    596,    180,     23,     1;
  6:  1312,   3256,   3032,   1328,    284,    28,    1;
  7:  4032,  11936,  13880,   8088,   2512,   412,   33,   1;
  8: 12240,  42144,  58944,  43200,  17916,  4260,  564,  38,  1;
  9: 36784, 144528, 236640, 209856, 109932, 34932, 6684, 740, 43, 1;
%F Formula
The row polynomials satisfy (6*n^2+6*n+(3*n^3+9*n^2+6*n)*x+(11*n^2+5*n)*x^2+(n^2+10*n)*x^3+n*x^4)*P(n,x) = -(18*n^3-6*n^2-72*n+24+(15*n^3+45*n^2-114*n-12)*x+(52*n^2-6*n-112)*x^2+(5*n^2+39*n-93)*x^3+(4*n-18)*x^4-x^5)*P(n-1,x) + (36*n^2+108*n+72+(72*n^2+348*n+348)*x+(39*n^2+393*n+630)*x^2+(3*n^2+176*n+527)*x^3+(24*n+197)*x^4+(n+25)*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[(-(18*n^3 - 6*n^2 - 72*n + 24 + (15*n^3 + 45*n^2 - 114*n - 12)*x + (52*n^2 - 6*n - 112)*x^2 + (5*n^2 + 39*n - 93)*x^3 + (4*n - 18)*x^4 - x^5)*P[n - 1, x] + (36*n^2 + 108*n + 72 + (72*n^2 + 348*n + 348)*x + (39*n^2 + 393*n + 630)*x^2 + (3*n^2 + 176*n + 527)*x^3 + (24*n + 197)*x^4 + (n + 25)*x^5 + x^6)*D[P[n - 1, x], x])/(6*n^2 + 6*n + (3*n^3 + 9*n^2 + 6*n)*x + (11*n^2 + 5*n)*x^2 + (n^2 + 10*n)*x^3 + n*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] = (-(18*n^3 - 6*n^2 - 72*n + 24 + (15*n^3 + 45*n^2 - 114*n - 12)*x + (52*n^2 - 6*n - 112)*x^2 + (5*n^2 + 39*n - 93)*x^3 + (4*n - 18)*x^4 - x^5)*v[i - 1] + (36*n^2 + 108*n + 72 + (72*n^2 + 348*n + 348)*x + (39*n^2 + 393*n + 630)*x^2 + (3*n^2 + 176*n + 527)*x^3 + (24*n + 197)*x^4 + (n + 25)*x^5 + x^6)*deriv(v[i - 1], x))/(6*n^2 + 6*n + (3*n^3 + 9*n^2 + 6*n)*x + (11*n^2 + 5*n)*x^2 + (n^2 + 10*n)*x^3 + n*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.