A133080 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2}\right)\,P_{n}(x) = \left(13 + 64\,x + 110\,x^{2} + 80\,x^{3} + 21\,x^{4} - 9\,n - 48\,n x - 90\,n x^{2} - 72\,n x^{3} - 21\,n x^{4}\right)\,P_{n-1}(x) + \left(3 + 25\,x + 78\,x^{2} + 114\,x^{3} + 79\,x^{4} + 21\,x^{5}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	1	6	9						16
3	2	16	42	36					96
4	1	12	54	108	81				256
5	2	28	156	432	594	324			1536
6	1	18	135	540	1215	1458	729		4096
7	2	40	342	1620	4590	7776	7290	2916	24576

All 15 rows

Technical details

Node id: n-35688ab8c7e1fea1 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-af66b97995ade75a

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 2,  4;
  2: 1,  6,   9;
  3: 2, 16,  42,   36;
  4: 1, 12,  54,  108,    81;
  5: 2, 28, 156,  432,   594,   324;
  6: 1, 18, 135,  540,  1215,  1458,   729;
  7: 2, 40, 342, 1620,  4590,  7776,  7290,   2916;
  8: 1, 24, 252, 1512,  5670, 13608, 20412,  17496,  6561;
  9: 2, 52, 600, 4032, 17388, 49896, 95256, 116640, 83106, 26244;

%F Formula

The row polynomials satisfy (2+4*x+2*x^2)*P(n,x) = -(9*n-13+(48*n-64)*x+(90*n-110)*x^2+(72*n-80)*x^3+(21*n-21)*x^4)*P(n-1,x) + (3+25*x+78*x^2+114*x^3+79*x^4+21*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[(-(9*n - 13 + (48*n - 64)*x + (90*n - 110)*x^2 + (72*n - 80)*x^3 + (21*n - 21)*x^4)*P[n - 1, x] + (3 + 25*x + 78*x^2 + 114*x^3 + 79*x^4 + 21*x^5)*D[P[n - 1, x], x])/(2 + 4*x + 2*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] = (-(9*n - 13 + (48*n - 64)*x + (90*n - 110)*x^2 + (72*n - 80)*x^3 + (21*n - 21)*x^4)*v[i - 1] + (3 + 25*x + 78*x^2 + 114*x^3 + 79*x^4 + 21*x^5)*deriv(v[i - 1], x))/(2 + 4*x + 2*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

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