Real-rooted atlas

A062190 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(5\,n + 10\,n x + 5\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(25 + 75\,x + 75\,x^{2} + 25\,x^{3} + 10\,n + 35\,n x + 40\,n x^{2} + 15\,n x^{3} + n^{2} + 4\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(5\,x + 15\,x^{2} + 15\,x^{3} + 5\,x^{4} + 2\,n x + 6\,n x^{2} + 6\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16713
2215636113
356252360165833
4126840198019807155641
5252231079201287010010300336365
646255442574060060750754804812376227305
77921201272072225225400400408408222768503881392065

All 15 rows

Technical details
Node id
n-9e4e89cef19362df · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-52731deea227dc21

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:    6,     7;
  2:   21,    56,     36;
  3:   56,   252,    360,     165;
  4:  126,   840,   1980,    1980,     715;
  5:  252,  2310,   7920,   12870,   10010,     3003;
  6:  462,  5544,  25740,   60060,   75075,    48048,    12376;
  7:  792, 12012,  72072,  225225,  400400,   408408,   222768,    50388;
  8: 1287, 24024, 180180,  720720, 1701700,  2450448,  2116296,  1007760,  203490;
  9: 2002, 45045, 411840, 2042040, 6126120, 11639628, 14108640, 10581480, 4476780, 817190;
%F Formula
The row polynomials satisfy (n^2+5*n+(2*n^2+10*n)*x+(n^2+5*n)*x^2)*P(n,x) = (n^2+10*n+25+(4*n^2+35*n+75)*x+(5*n^2+40*n+75)*x^2+(2*n^2+15*n+25)*x^3)*P(n-1,x) + ((2*n+5)*x+(6*n+15)*x^2+(6*n+15)*x^3+(2*n+5)*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^2 + 10*n + 25 + (4*n^2 + 35*n + 75)*x + (5*n^2 + 40*n + 75)*x^2 + (2*n^2 + 15*n + 25)*x^3)*P[n - 1, x] + ((2*n + 5)*x + (6*n + 15)*x^2 + (6*n + 15)*x^3 + (2*n + 5)*x^4)*D[P[n - 1, x], x])/(n^2 + 5*n + (2*n^2 + 10*n)*x + (n^2 + 5*n)*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^2 + 10*n + 25 + (4*n^2 + 35*n + 75)*x + (5*n^2 + 40*n + 75)*x^2 + (2*n^2 + 15*n + 25)*x^3)*v[i - 1] + ((2*n + 5)*x + (6*n + 15)*x^2 + (6*n + 15)*x^3 + (2*n + 5)*x^4)*deriv(v[i - 1], x))/(n^2 + 5*n + (2*n^2 + 10*n)*x + (n^2 + 5*n)*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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.