Real-rooted atlas

A063967 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 3\,x - 3\,x^{2} - x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(-1 - 13\,x - 57\,x^{2} - 103\,x^{3} - 82\,x^{4} - 24\,x^{5} + n + 9\,n x + 33\,n x^{2} + 55\,n x^{3} + 42\,n x^{4} + 12\,n x^{5}\right)\,P_{n-1}(x) + \left(-x - 11\,x^{2} - 39\,x^{3} - 61\,x^{4} - 44\,x^{5} - 12\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2123
315612
418201645
5111437244171
611475194244120648
71171164097907963282457
81201667441952300825288969315
9123225122640818520109207872244835316
1012629318827609201223487238256241446688133893
1112937027391304941873917981359601303687316818272507627
1213245638242099579344210884394672510288434560219584499201924560
131355515164321221398614375439919841619188185742414226086536961363847296561
1413865567864718623274883859922366424419596639543265925924587616193286437260827663363
1514176887176702436957015082004631105107599501883778024479256229102561460700856821121017984104879772
Technical details
Node id
n-bcafa7bf502f610d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-bf438b133f65bda8

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,  2;
   3: 1,  5,   6;
   4: 1,  8,  20,   16;
   5: 1, 11,  43,   72,   44;
   6: 1, 14,  75,  194,  244,   120;
   7: 1, 17, 116,  409,  790,   796,   328;
   8: 1, 20, 166,  744, 1952,  3008,  2528,   896;
   9: 1, 23, 225, 1226, 4081,  8520, 10920,  7872,  2448;
  10: 1, 26, 293, 1882, 7609, 20122, 34872, 38256, 24144, 6688;
%F Formula
The row polynomials satisfy (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (n-1+(9*n-13)*x+(33*n-57)*x^2+(55*n-103)*x^3+(42*n-82)*x^4+(12*n-24)*x^5)*P(n-1,x) - (x+11*x^2+39*x^3+61*x^4+44*x^5+12*x^6)*d/dx 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[((n - 1 + (9*n - 13)*x + (33*n - 57)*x^2 + (55*n - 103)*x^3 + (42*n - 82)*x^4 + (12*n - 24)*x^5)*P[n - 1, x] - (x + 11*x^2 + 39*x^3 + 61*x^4 + 44*x^5 + 12*x^6)*D[P[n - 1, x], x])/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*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] = ((n - 1 + (9*n - 13)*x + (33*n - 57)*x^2 + (55*n - 103)*x^3 + (42*n - 82)*x^4 + (12*n - 24)*x^5)*v[i - 1] - (x + 11*x^2 + 39*x^3 + 61*x^4 + 44*x^5 + 12*x^6)*deriv(v[i - 1], x))/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*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

Candidate equalities ?

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741