Real-rooted atlas

A039756 → 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

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-2 - 13\,x - 28\,x^{2} - 25\,x^{3} - 8\,x^{4} + 4\,n + 20\,n x + 36\,n x^{2} + 28\,n x^{3} + 8\,n x^{4}\right)\,P_{n-1}(x) + \left(-2 - 14\,x - 38\,x^{2} - 50\,x^{3} - 32\,x^{4} - 8\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
26181337
32411016479377
4116720164816446014729
56485084157522405618078533968957
6408838824152036313928360094217326538611140157
7286403196401515488395404461266325633150284360060752721028721

All 15 rows

Technical details
Node id
n-75e4d4a68031f00a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-6e298d05f674ef99

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:       2,        3;
  2:       6,       18,        13;
  3:      24,      110,       164,        79;
  4:     116,      720,      1648,      1644,        601;
  5:     648,     5084,     15752,     24056,      18078,       5339;
  6:    4088,    38824,    152036,    313928,     360094,     217326,      53861;
  7:   28640,   319640,   1515488,   3954044,    6126632,    5633150,    2843600,     607527;
  8:  219920,  2822528,  15728896,  49681504,   97229416,  120655744,   92657840,   40236920,   7560625;
  9: 1832224, 26597168, 170460224, 632788288, 1498809776, 2347903736, 2431379728, 1604212256, 611656730, 102637235;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (4*n-2+(20*n-13)*x+(36*n-28)*x^2+(28*n-25)*x^3+(8*n-8)*x^4)*P(n-1,x) - (2+14*x+38*x^2+50*x^3+32*x^4+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 + (20*n - 13)*x + (36*n - 28)*x^2 + (28*n - 25)*x^3 + (8*n - 8)*x^4)*P[n - 1, x] - (2 + 14*x + 38*x^2 + 50*x^3 + 32*x^4 + 8*x^5)*D[P[n - 1, x], x])/(1 + 2*x + 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 + (20*n - 13)*x + (36*n - 28)*x^2 + (28*n - 25)*x^3 + (8*n - 8)*x^4)*v[i - 1] - (2 + 14*x + 38*x^2 + 50*x^3 + 32*x^4 + 8*x^5)*deriv(v[i - 1], x))/(1 + 2*x + 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.