Real-rooted atlas

A008278 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 3\,x - 3\,x^{2} - x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(-n - n x + 7\,n x^{2} + 17\,n x^{3} + 14\,n x^{4} + 4\,n x^{5} - 2\,n^{2} x - 6\,n^{2} x^{2} - 3\,n^{2} x^{3} + 7\,n^{2} x^{4} + 9\,n^{2} x^{5} + 3\,n^{2} x^{6} - n^{3} x^{2} - 4\,n^{3} x^{3} - 6\,n^{3} x^{4} - 4\,n^{3} x^{5} - n^{3} x^{6}\right)\,P_{n-2}(x) + \left(n x^{2} + 4\,n x^{3} + 6\,n x^{4} + 4\,n x^{5} + n x^{6} + n^{2} x^{3} + 4\,n^{2} x^{4} + 6\,n^{2} x^{5} + 4\,n^{2} x^{6} + n^{2} x^{7}\right)\,P_{n-2}'(x) + \left(-1 - 2\,x + 2\,x^{2} + 8\,x^{3} + 7\,x^{4} + 2\,x^{5} + 2\,n + 4\,n x - 3\,n x^{2} - 13\,n x^{3} - 11\,n x^{4} - 3\,n x^{5} + 2\,n^{2} x + 7\,n^{2} x^{2} + 9\,n^{2} x^{3} + 5\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}(x) + \left(x^{2} + 4\,x^{3} + 6\,x^{4} + 4\,x^{5} + x^{6} - n x^{2} - 4\,n x^{3} - 6\,n x^{4} - 4\,n x^{5} - n x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2134
3181019
41154838102
5124141274161601
6135325113515857473828
7148645352086609484375326111
81631155906534440653525912920253189458

All 15 rows

Technical details
Node id
n-abf91611a40cf61a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7b7fb3e3d2809593

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,  3;
   3: 1,  8,   10;
   4: 1, 15,   48,    38;
   5: 1, 24,  141,   274,    161;
   6: 1, 35,  325,  1135,   1585,     747;
   7: 1, 48,  645,  3520,   8660,    9484,    3753;
   8: 1, 63, 1155,  9065,  34440,   65352,   59129,   20253;
   9: 1, 80, 1918, 20468, 111055,  321776,  497735,  385018,  116642;
  10: 1, 99, 3006, 41874, 307671, 1263801, 2954959, 3865701, 2618958, 713130;
%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+(2*n^2+n)*x+(n^3+6*n^2-7*n)*x^2+(4*n^3+3*n^2-17*n)*x^3+(6*n^3-7*n^2-14*n)*x^4+(4*n^3-9*n^2-4*n)*x^5+(n^3-3*n^2)*x^6)*P(n-2,x) + (n*x^2+(n^2+4*n)*x^3+(4*n^2+6*n)*x^4+(6*n^2+4*n)*x^5+(4*n^2+n)*x^6+n^2*x^7)*d/dx P(n-2,x) + (2*n-1+(2*n^2+4*n-2)*x+(7*n^2-3*n+2)*x^2+(9*n^2-13*n+8)*x^3+(5*n^2-11*n+7)*x^4+(n^2-3*n+2)*x^5)*P(n-1,x) - ((n-1)*x^2+(4*n-4)*x^3+(6*n-6)*x^4+(4*n-4)*x^5+(n-1)*x^6)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + (2*n^2 + n)*x + (n^3 + 6*n^2 - 7*n)*x^2 + (4*n^3 + 3*n^2 - 17*n)*x^3 + (6*n^3 - 7*n^2 - 14*n)*x^4 + (4*n^3 - 9*n^2 - 4*n)*x^5 + (n^3 - 3*n^2)*x^6)*P[n - 2, x] + (n*x^2 + (n^2 + 4*n)*x^3 + (4*n^2 + 6*n)*x^4 + (6*n^2 + 4*n)*x^5 + (4*n^2 + n)*x^6 + n^2*x^7)*D[P[n - 2, x], x] + (2*n - 1 + (2*n^2 + 4*n - 2)*x + (7*n^2 - 3*n + 2)*x^2 + (9*n^2 - 13*n + 8)*x^3 + (5*n^2 - 11*n + 7)*x^4 + (n^2 - 3*n + 2)*x^5)*P[n - 1, x] - ((n - 1)*x^2 + (4*n - 4)*x^3 + (6*n - 6)*x^4 + (4*n - 4)*x^5 + (n - 1)*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; v[2] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = (-(n + (2*n^2 + n)*x + (n^3 + 6*n^2 - 7*n)*x^2 + (4*n^3 + 3*n^2 - 17*n)*x^3 + (6*n^3 - 7*n^2 - 14*n)*x^4 + (4*n^3 - 9*n^2 - 4*n)*x^5 + (n^3 - 3*n^2)*x^6)*v[i - 2] + (n*x^2 + (n^2 + 4*n)*x^3 + (4*n^2 + 6*n)*x^4 + (6*n^2 + 4*n)*x^5 + (4*n^2 + n)*x^6 + n^2*x^7)*deriv(v[i - 2], x) + (2*n - 1 + (2*n^2 + 4*n - 2)*x + (7*n^2 - 3*n + 2)*x^2 + (9*n^2 - 13*n + 8)*x^3 + (5*n^2 - 11*n + 7)*x^4 + (n^2 - 3*n + 2)*x^5)*v[i - 1] - ((n - 1)*x^2 + (4*n - 4)*x^3 + (6*n - 6)*x^4 + (4*n - 4)*x^5 + (n - 1)*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

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.