A106828: Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 0 and 0 <= k <= floor(n/2)).

A106828 ↗ nonnegative coefficients

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

Recurrence from A106828

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1									0
2	0	1							1
3	0	2							2
4	0	6	3						9
5	0	24	20						44
6	0	120	130	15					265
7	0	720	924	210					1854
8	0	5040	7308	2380	105				14833
9	0	40320	64224	26432	2520				133496
10	0	362880	623376	303660	44100	945			1334961
11	0	3628800	6636960	3678840	705320	34650			14684570
12	0	39916800	76998240	47324376	11098780	866250	10395		176214841
13	0	479001600	967524480	647536032	177331440	18858840	540540		2290792932
14	0	6227020800	13096736640	9418945536	2920525608	389449060	18288270	135135	32071101049

b-files: A000166 (row sums), A008306 ?

Technical details

Node id: n-fd92598b7a526ebf · revision 0 · JSON
Definition source: oeis_catalog:A106828
Fingerprint: f6-adb63b75c11d63d7

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 ?

A008306 triangle 36 terms, 11 rows · tabf · nonn · catalog b-file?

Row sums ? A257953, A260216, A000166, A260091, A260115

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,     3;
  5: 0,    24,    20;
  6: 0,   120,   130,    15;
  7: 0,   720,   924,   210;
  8: 0,  5040,  7308,  2380,  105;
  9: 0, 40320, 64224, 26432, 2520;

%F Formula

The row polynomials satisfy P(n,x) = (n-1)*P(n-1,x) + (n-1)*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] + (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] = 0; for(n = 2, N, my(i = n + 1); v[i] = (n - 1)*v[i - 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

A008306 triangle, offset 2, from P₂, low zeros dropped

%e Example

Triangle begins:
   2:       1;
   3:       2;
   4:       6,       3;
   5:      24,      20;
   6:     120,     130,      15;
   7:     720,     924,     210;
   8:    5040,    7308,    2380,    105;
   9:   40320,   64224,   26432,   2520;
  10:  362880,  623376,  303660,  44100,   945;
  11: 3628800, 6636960, 3678840, 705320, 34650;

%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-1)*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 4, with P(2,x) = x, P(3,x) = 2*x.

%t Mathematica

P[2, x_] := x;
P[3, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[(n - 1)*P[n - 1, x] + (n - 1)*x*P[n - 2, x]];
Table[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], {n, 2, 11}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N - 1)); v[1] = x; v[2] = 2*x; for(n = 4, N, my(i = n - 1); v[i] = (n - 1)*v[i - 1] + (n - 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 + 1, v[i])))};
T(11)

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 ?

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