A181289: Triangle read by rows: T(n,k) is the number of 2-compositions of n having length k (0 <= k <= n).

A181289 ↗ nonnegative coefficients

Proved for all n all roots real, in [−1, ∞)interlacing (weak)

Recurrence from A181289

\(\left(-2 + n\right)\,P_{n}(x) = \left(-4 - 4\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(2 + 2\,x - n - n x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 2\,x\)

\(P_{2}(x) = 3\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	2														2
2	0	3	4													7
3	0	4	12	8												24
4	0	5	25	36	16											82
5	0	6	44	102	96	32										280
6	0	7	70	231	344	240	64									956
7	0	8	104	456	952	1040	576	128								3264
8	0	9	147	819	2241	3400	2928	1344	256							11144
9	0	10	200	1372	4712	9290	11040	7840	3072	512						38048
10	0	11	264	2178	9108	22363	34332	33488	20224	6912	1024					129904
11	0	12	340	3312	16488	48940	93060	116760	96512	50688	15360	2048				443520
12	0	13	429	4862	28314	99385	227305	351820	372832	267264	124160	33792	4096			1514272
13	0	14	532	6930	46552	189970	511380	948430	1236032	1132992	716800	298496	73728	8192		5170048
14	0	15	650	9633	73788	345345	1076010	2340495	3644272	4097952	3308160	1872640	706560	159744	16384	17651648

b-files: A003480 (row sums), A020727 (row sums), A181289 ?

Technical details

Node id: n-4ad294e2bfbb52a6 · revision 0 · JSON
Definition source: oeis_catalog:A181289
Fingerprint: f6-fd2eb6e3f0c1aca6

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): no

Matches ?

A181289 triangle 61 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A003480, A020727

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0,  2;
  2: 0,  3,   4;
  3: 0,  4,  12,    8;
  4: 0,  5,  25,   36,   16;
  5: 0,  6,  44,  102,   96,   32;
  6: 0,  7,  70,  231,  344,  240,    64;
  7: 0,  8, 104,  456,  952, 1040,   576,  128;
  8: 0,  9, 147,  819, 2241, 3400,  2928, 1344,  256;
  9: 0, 10, 200, 1372, 4712, 9290, 11040, 7840, 3072, 512;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) - (n-2+(n-2)*x)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2*x, P(2,x) = 3*x + 4*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[2, x_] := 3*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (n - 2 + (n - 2)*x)*P[n - 2, x])/(n - 2)]];
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] = 2*x; v[3] = 3*x + 4*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] - (n - 2 + (n - 2)*x)*v[i - 2])/(n - 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

A181289 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0,  2;
  2: 0,  3,   4;
  3: 0,  4,  12,    8;
  4: 0,  5,  25,   36,   16;
  5: 0,  6,  44,  102,   96,   32;
  6: 0,  7,  70,  231,  344,  240,    64;
  7: 0,  8, 104,  456,  952, 1040,   576,  128;
  8: 0,  9, 147,  819, 2241, 3400,  2928, 1344,  256;
  9: 0, 10, 200, 1372, 4712, 9290, 11040, 7840, 3072, 512;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) - (n-2+(n-2)*x)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2*x, P(2,x) = 3*x + 4*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[2, x_] := 3*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (n - 2 + (n - 2)*x)*P[n - 2, x])/(n - 2)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; v[3] = 3*x + 4*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] - (n - 2 + (n - 2)*x)*v[i - 2])/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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, +∞)
by 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 ≥ 2
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