A199478: Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.

A199478 ↗ nonnegative coefficients

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

Recurrence from A199478

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

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

\(P_{1}(x) = 1 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	1	1								2
1	1	3								4
2	1	5	2							8
3	1	7	8							16
4	1	9	18	4						32
5	1	11	32	20						64
6	1	13	50	56	8					128
7	1	15	72	120	48					256
8	1	17	98	220	160	16				512
9	1	19	128	364	400	112				1024
10	1	21	162	560	840	432	32			2048
11	1	23	200	816	1568	1232	256			4096
12	1	25	242	1140	2688	2912	1120	64		8192
13	1	27	288	1540	4320	6048	3584	576		16384
14	1	29	338	2024	6600	11424	9408	2816	128	32768

b-files: A199478 ?

Technical details

Node id: n-ff9680a2cfd23e93 · revision 0 · JSON
Definition source: oeis_catalog:A199478
Fingerprint: f6-1baf5f9ce9075389

Observed on the stored rows ?

Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
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 ?

A199478 triangle 64 terms, 13 rows · tabf · nonn · catalog b-file?

Row sums ? A050732, A090129, A155559, A011782, A034008, A118655, A131577, A000079

Submission text ?

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

%e Example

Triangle begins:
  0: 1,  1;
  1: 1,  3;
  2: 1,  5,   2;
  3: 1,  7,   8;
  4: 1,  9,  18,   4;
  5: 1, 11,  32,  20;
  6: 1, 13,  50,  56,   8;
  7: 1, 15,  72, 120,  48;
  8: 1, 17,  98, 220, 160,  16;
  9: 1, 19, 128, 364, 400, 112;

%F Formula

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

%t Mathematica

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

A199478 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1,  1;
   2: 1,  3;
   3: 1,  5,   2;
   4: 1,  7,   8;
   5: 1,  9,  18,   4;
   6: 1, 11,  32,  20;
   7: 1, 13,  50,  56,   8;
   8: 1, 15,  72, 120,  48;
   9: 1, 17,  98, 220, 160,  16;
  10: 1, 19, 128, 364, 400, 112;

%F Formula

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

%t Mathematica

P[1, x_] := 1 + x;
P[2, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + 2*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + x; v[2] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = v[i - 1] + 2*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, v[i])))};
T(10)

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

Used in 10

2 more

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