Real-rooted atlas

A062264 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 8\,n x + 4\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(12 + 44\,x + 44\,x^{2} + 4\,x^{3} - 8\,x^{4} + 14\,n + 54\,n x + 70\,n x^{2} + 34\,n x^{3} + 4\,n x^{4} + 4\,n^{2} + 17\,n^{2} x + 26\,n^{2} x^{2} + 17\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-4\,x - 20\,x^{2} - 36\,x^{3} - 28\,x^{4} - 8\,x^{5} - 2\,n x - 10\,n x^{2} - 18\,n x^{3} - 14\,n x^{4} - 4\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
161117
2289885211
312061210325752339
4495330081908964364924598
520021644553680870107001022363251510
68008780783153156749608075105118301342452529946
7318243589041725360458230572599806860490357966079545525193978

All 15 rows

Technical details
Node id
n-cd0853159b2e16c3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5f88e8a070faa8e0

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,      11;
  2:     28,      98,       85;
  3:    120,     612,     1032,       575;
  4:    495,    3300,     8190,      8964,      3649;
  5:   2002,   16445,    53680,     87010,     70010,     22363;
  6:   8008,   78078,   315315,    674960,    807510,    511830,    134245;
  7:  31824,  358904,  1725360,   4582305,   7259980,   6860490,   3579660,    795455;
  8: 125970, 1612416,  8984976,  28464800,  56066920,  70299320,  54787460,  24261640,   4673345;
  9: 497420, 7122150, 45116640, 165939312, 390478032, 609570360, 631223040, 418054140, 160668270, 27298155;
%F Formula
The row polynomials satisfy (n^2+4*n+(2*n^2+8*n)*x+(n^2+4*n)*x^2)*P(n,x) = (4*n^2+14*n+12+(17*n^2+54*n+44)*x+(26*n^2+70*n+44)*x^2+(17*n^2+34*n+4)*x^3+(4*n^2+4*n-8)*x^4)*P(n-1,x) - ((2*n+4)*x+(10*n+20)*x^2+(18*n+36)*x^3+(14*n+28)*x^4+(4*n+8)*x^5)*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[((4*n^2 + 14*n + 12 + (17*n^2 + 54*n + 44)*x + (26*n^2 + 70*n + 44)*x^2 + (17*n^2 + 34*n + 4)*x^3 + (4*n^2 + 4*n - 8)*x^4)*P[n - 1, x] - ((2*n + 4)*x + (10*n + 20)*x^2 + (18*n + 36)*x^3 + (14*n + 28)*x^4 + (4*n + 8)*x^5)*D[P[n - 1, x], x])/(n^2 + 4*n + (2*n^2 + 8*n)*x + (n^2 + 4*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] = ((4*n^2 + 14*n + 12 + (17*n^2 + 54*n + 44)*x + (26*n^2 + 70*n + 44)*x^2 + (17*n^2 + 34*n + 4)*x^3 + (4*n^2 + 4*n - 8)*x^4)*v[i - 1] - ((2*n + 4)*x + (10*n + 20)*x^2 + (18*n + 36)*x^3 + (14*n + 28)*x^4 + (4*n + 8)*x^5)*deriv(v[i - 1], x))/(n^2 + 4*n + (2*n^2 + 8*n)*x + (n^2 + 4*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

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.