A330609: T(n, k) = binomial(n-k-1, k-1)*(n-k)!/k! for n >= 0 and 0 <= k <= floor(n/2). Irregular triangle read by rows.

A330609 ↗ nonnegative coefficients

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

Recurrence from A330609

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1					0
2	0	1			1
3	0	2			2
4	0	6	1		7
5	0	24	6		30
6	0	120	36	1	157
7	0	720	240	12	972

All 15 rows b-files: A001053 (row sums), A221913 ?

Technical details

Node id: n-745333beb735b03d · revision 0 · JSON
Definition source: oeis_catalog:A330609
Fingerprint: f6-0dfc3348a5ee8b79

Observed on the stored rows ?

Degrees: 0, -1, 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 ?

A221913 triangle 42 terms, 12 rows · tabf · nonn b-file?

Row sums ? A001053

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0;
  2: 0,     1;
  3: 0,     2;
  4: 0,     6,     1;
  5: 0,    24,     6;
  6: 0,   120,    36,    1;
  7: 0,   720,   240,   12;
  8: 0,  5040,  1800,  120,  1;
  9: 0, 40320, 15120, 1200, 20;

%F Formula

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

%t Mathematica

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

A221913 triangle, offset 1, from P₂, low zeros dropped

%e Example

Triangle begins:
   1:       1;
   2:       2;
   3:       6,       1;
   4:      24,       6;
   5:     120,      36,      1;
   6:     720,     240,     12;
   7:    5040,    1800,    120,    1;
   8:   40320,   15120,   1200,   20;
   9:  362880,  141120,  12600,  300,  1;
  10: 3628800, 1451520, 141120, 4200, 30;

%F Formula

Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = n*P(n-1,x) + x*P(n-2,x) for n >= 3, with P(1,x) = x, P(2,x) = 2*x.

%t Mathematica

P[1, x_] := x;
P[2, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[n*P[n - 1, x] + x*P[n - 2, x]];
Table[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = x; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = n*v[i - 1] + x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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 ≥ 2, 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 ≥ 2: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 2, 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 ≥ 3
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

Related ?

Same fingerprint ?

A180047: Coefficient triangle of the numerators of the (n-th convergents to) the continued fraction w/(1 + w/(2 + w/(3 …

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