A136394: Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).

A136394 ↗ nonnegative coefficients

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

Recurrence from A136394

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	1	1			2
3	1	5			6
4	1	20	3		24
5	1	84	35		120
6	1	409	295	15	720
7	1	2365	2359	315	5040

All 15 rows b-files: A000142 (row sums), A136394 ?

Technical details

Node id: n-497753d6192915db · revision 0 · JSON
Definition source: oeis_catalog:A136394
Fingerprint: f6-36b7427135e5b844

Observed on the stored rows ?

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

A136394 triangle 46 terms, 12 rows · tabf · nonn · catalog b-file?

Row sums ? A370360, A000142, A159333, A358185, A338112, A104150, A124355, A354074

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 1,      1;
  3: 1,      5;
  4: 1,     20,      3;
  5: 1,     84,     35;
  6: 1,    409,    295,    15;
  7: 1,   2365,   2359,   315;
  8: 1,  16064,  19670,  4480,  105;
  9: 1, 125664, 177078, 56672, 3465;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[n*P[n - 1, x] - (n - 1 - (n - 1)*x)*P[n - 2, 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; for(n = 2, N, my(i = n + 1); v[i] = n*v[i - 1] - (n - 1 - (n - 1)*x)*v[i - 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

A136394 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 1,      1;
  3: 1,      5;
  4: 1,     20,      3;
  5: 1,     84,     35;
  6: 1,    409,    295,    15;
  7: 1,   2365,   2359,   315;
  8: 1,  16064,  19670,  4480,  105;
  9: 1, 125664, 177078, 56672, 3465;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[n*P[n - 1, x] - (n - 1 - (n - 1)*x)*P[n - 2, 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; for(n = 2, N, my(i = n + 1); v[i] = n*v[i - 1] - (n - 1 - (n - 1)*x)*v[i - 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 (−∞, 0)
by Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[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