A305540: Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.

A305540 ↗ nonnegative coefficients

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

Recurrence from A305540

\(P_{n}(x) = \left(1 + 2\,x\right)\,P_{n-2}(x) + \left(x + x^{2}\right)\,P_{n-2}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	2							3
3	1	4	3						8
4	1	6	6						13
5	1	10	21	12					44
6	1	14	36	24					75
7	1	22	93	132	60				308
8	1	30	150	240	120				541
9	1	46	345	900	960	360			2612
10	1	62	540	1560	1800	720			4683
11	1	94	1173	4980	9300	7920	2520		25988
12	1	126	1806	8400	16800	15120	5040		47293
13	1	190	3801	24612	71400	103320	73080	20160	296564
14	1	254	5796	40824	126000	191520	141120	40320	545835

b-files: A305540 ?

Technical details

Node id: n-f1291d4c6c67167e · revision 0 · JSON
Definition source: oeis_catalog:A305540
Fingerprint: f6-fdab596c99811270

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 ?

A305540 triangle 71 terms, 15 rows · tabf · nonn · catalog b-file?

Row sums ? A327868

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  2;
  3: 1,  4,   3;
  4: 1,  6,   6;
  5: 1, 10,  21,  12;
  6: 1, 14,  36,  24;
  7: 1, 22,  93, 132,  60;
  8: 1, 30, 150, 240, 120;
  9: 1, 46, 345, 900, 960, 360;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-2,x) + (x+x^2)*d/dx P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 2, x] + (x + x^2)*D[P[n - 2, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 2] + (x + x^2)*deriv(v[i - 2], x)); 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

A305540 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,  1;
   3: 1,  2;
   4: 1,  4,   3;
   5: 1,  6,   6;
   6: 1, 10,  21,  12;
   7: 1, 14,  36,  24;
   8: 1, 22,  93, 132,  60;
   9: 1, 30, 150, 240, 120;
  10: 1, 46, 345, 900, 960, 360;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-2,x) + (x+x^2)*d/dx P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 2, x] + (x + x^2)*D[P[n - 2, x], x]];
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 + x; for(n = 3, N, my(i = n); v[i] = (1 + 2*x)*v[i - 2] + (x + x^2)*deriv(v[i - 2], x)); 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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Iterated real-rootedness preservers
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers

Proof chains ? JSON

Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Used in 5

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.