Triangle T(n,k), read by rows, given by (1,-1,0,0,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

A199881 (tail) ↗ nonnegative coefficients

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

Recurrence from A199881

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

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

\(P_{1}(x) = x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	1	1															2
1	0	1	1														2
2	0	1	2	1													4
3	0	0	2	3	1												6
4	0	0	1	4	4	1											10
5	0	0	0	3	7	5	1										16
6	0	0	0	1	7	11	6	1									26
7	0	0	0	0	4	14	16	7	1								42
8	0	0	0	0	1	11	25	22	8	1							68
9	0	0	0	0	0	5	25	41	29	9	1						110
10	0	0	0	0	0	1	16	50	63	37	10	1					178
11	0	0	0	0	0	0	6	41	91	92	46	11	1				288
12	0	0	0	0	0	0	1	22	91	154	129	56	12	1			466
13	0	0	0	0	0	0	0	7	63	182	246	175	67	13	1		754
14	0	0	0	0	0	0	0	1	29	154	336	375	231	79	14	1	1220

b-files: A078642 (row sums) ?

Technical details

Node id: n-f239c3b0ba932394 · revision 0 · JSON
Definition source: oeis_catalog:A199881
Fingerprint: f6-1cfd4c48d5eaa881

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 ?

Row sums ? A128588, A163733, A055389, A078642, A198834, A006355, A068922, A118658

Submission text ?

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

%e Example

Triangle begins:
  0: 1, 1;
  1: 0, 1, 1;
  2: 0, 1, 2, 1;
  3: 0, 0, 2, 3, 1;
  4: 0, 0, 1, 4, 4,  1;
  5: 0, 0, 0, 3, 7,  5,  1;
  6: 0, 0, 0, 1, 7, 11,  6,  1;
  7: 0, 0, 0, 0, 4, 14, 16,  7,  1;
  8: 0, 0, 0, 0, 1, 11, 25, 22,  8, 1;
  9: 0, 0, 0, 0, 0,  5, 25, 41, 29, 9, 1;

%F Formula

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

%t Mathematica

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

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