Real-rooted atlas

A008288 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-2 - 6\,x - 6\,x^{2} - 2\,x^{3} + 2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(58 + 468\,x + 1510\,x^{2} + 2512\,x^{3} + 2286\,x^{4} + 1084\,x^{5} + 210\,x^{6} - 45\,n - 354\,n x - 1149\,n x^{2} - 1948\,n x^{3} - 1815\,n x^{4} - 882\,n x^{5} - 175\,n x^{6} + 9\,n^{2} + 66\,n^{2} x + 209\,n^{2} x^{2} + 356\,n^{2} x^{3} + 339\,n^{2} x^{4} + 170\,n^{2} x^{5} + 35\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(20 + 236\,x + 1082\,x^{2} + 2578\,x^{3} + 3512\,x^{4} + 2768\,x^{5} + 1178\,x^{6} + 210\,x^{7} - 6\,n - 70\,n x - 322\,n x^{2} - 778\,n x^{3} - 1082\,n x^{4} - 874\,n x^{5} - 382\,n x^{6} - 70\,n x^{7}\right)\,P_{n-1}'(x) + \left(1 + 16\,x + 102\,x^{2} + 340\,x^{3} + 660\,x^{4} + 776\,x^{5} + 546\,x^{6} + 212\,x^{7} + 35\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2268
35304176
412108308276704
5293481510279618616544
6701050612417316236941254660800
71693042222918497217747119240284581564928
84088568756083630001022984169092015173205702165249024

All 15 rows

Technical details
Node id
n-70b12dfbbd98514b · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-07ec67562a37d493

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:    2,     6;
   3:    5,    30,     41;
   4:   12,   108,    308,     276;
   5:   29,   348,   1510,    2796,     1861;
   6:   70,  1050,   6124,   17316,    23694,    12546;
   7:  169,  3042,  22291,   84972,   177471,   192402,     84581;
   8:  408,  8568,  75608,  363000,  1022984,  1690920,   1517320,    570216;
   9:  985, 23640, 244012, 1413576,  5022726, 11201832,  15306508,  11713848,  3844201;
  10: 2378, 64206, 758920, 5150712, 22105932, 62183844, 114601768, 133395480, 88979386, 25916286;
%F Formula
The row polynomials satisfy (2*n-2+(6*n-6)*x+(6*n-6)*x^2+(2*n-2)*x^3)*P(n,x) = (9*n^2-45*n+58+(66*n^2-354*n+468)*x+(209*n^2-1149*n+1510)*x^2+(356*n^2-1948*n+2512)*x^3+(339*n^2-1815*n+2286)*x^4+(170*n^2-882*n+1084)*x^5+(35*n^2-175*n+210)*x^6)*P(n-1,x) - (6*n-20+(70*n-236)*x+(322*n-1082)*x^2+(778*n-2578)*x^3+(1082*n-3512)*x^4+(874*n-2768)*x^5+(382*n-1178)*x^6+(70*n-210)*x^7)*d/dx P(n-1,x) + (1+16*x+102*x^2+340*x^3+660*x^4+776*x^5+546*x^6+212*x^7+35*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 45*n + 58 + (66*n^2 - 354*n + 468)*x + (209*n^2 - 1149*n + 1510)*x^2 + (356*n^2 - 1948*n + 2512)*x^3 + (339*n^2 - 1815*n + 2286)*x^4 + (170*n^2 - 882*n + 1084)*x^5 + (35*n^2 - 175*n + 210)*x^6)*P[n - 1, x] - (6*n - 20 + (70*n - 236)*x + (322*n - 1082)*x^2 + (778*n - 2578)*x^3 + (1082*n - 3512)*x^4 + (874*n - 2768)*x^5 + (382*n - 1178)*x^6 + (70*n - 210)*x^7)*D[P[n - 1, x], x] + (1 + 16*x + 102*x^2 + 340*x^3 + 660*x^4 + 776*x^5 + 546*x^6 + 212*x^7 + 35*x^8)*D[P[n - 1, x], {x, 2}])/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((9*n^2 - 45*n + 58 + (66*n^2 - 354*n + 468)*x + (209*n^2 - 1149*n + 1510)*x^2 + (356*n^2 - 1948*n + 2512)*x^3 + (339*n^2 - 1815*n + 2286)*x^4 + (170*n^2 - 882*n + 1084)*x^5 + (35*n^2 - 175*n + 210)*x^6)*v[i - 1] - (6*n - 20 + (70*n - 236)*x + (322*n - 1082)*x^2 + (778*n - 2578)*x^3 + (1082*n - 3512)*x^4 + (874*n - 2768)*x^5 + (382*n - 1178)*x^6 + (70*n - 210)*x^7)*deriv(v[i - 1], x) + (1 + 16*x + 102*x^2 + 340*x^3 + 660*x^4 + 776*x^5 + 546*x^6 + 212*x^7 + 35*x^8)*deriv(deriv(v[i - 1], x), x))/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*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

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