A308737: Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is the 1-tiered binomial coefficient.

A308737 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A308737

\(P_{n}(x) = \left(3 + 3\,x\right)\,P_{n-1}(x) + \left(-2 - 4\,x - 2\,x^{2}\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	3														4
2	1	8	7													16
3	1	17	31	15												64
4	1	34	96	94	31											256
5	1	67	258	382	253	63										1024
6	1	132	645	1280	1275	636	127									4096
7	1	261	1545	3845	5115	3831	1531	255								16384
8	1	518	3598	10766	17920	17906	10738	3578	511							65536
9	1	1031	8212	28700	57358	71666	57316	28652	8185	1023						262144
10	1	2056	18459	73776	172074	258048	258006	171984	73701	18424	2047					1048576
11	1	4105	40995	184395	491610	860202	1032150	860070	491445	184285	40951	4095				4194304
12	1	8202	90156	450670	1351845	2703492	3784704	3784572	2703195	1351570	450516	90102	8191			16777216
13	1	16395	196662	1081498	3604755	8110377	12976260	15138684	12975831	8109805	3604326	1081290	196597	16383		67108864
14	1	32780	426049	2556112	9372077	23429692	42172845	56229888	56229459	42171844	23428691	9371440	2555839	425972	32767	268435456

b-files: A000302 (row sums), A308737 ?

Technical details

Node id: n-13e23eca5f474574 · revision 0 · JSON
Definition source: oeis_catalog:A308737
Fingerprint: f6-6fbba7c3cf69e7d1

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): yes

Matches ?

A308737 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A155543, A347478, A000302

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,    3;
  2: 1,    8,    7;
  3: 1,   17,   31,    15;
  4: 1,   34,   96,    94,    31;
  5: 1,   67,  258,   382,   253,    63;
  6: 1,  132,  645,  1280,  1275,   636,   127;
  7: 1,  261, 1545,  3845,  5115,  3831,  1531,   255;
  8: 1,  518, 3598, 10766, 17920, 17906, 10738,  3578,  511;
  9: 1, 1031, 8212, 28700, 57358, 71666, 57316, 28652, 8185, 1023;

%F Formula

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

%t Mathematica

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

A308737 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,    3;
  2: 1,    8,    7;
  3: 1,   17,   31,    15;
  4: 1,   34,   96,    94,    31;
  5: 1,   67,  258,   382,   253,    63;
  6: 1,  132,  645,  1280,  1275,   636,   127;
  7: 1,  261, 1545,  3845,  5115,  3831,  1531,   255;
  8: 1,  518, 3598, 10766, 17920, 17906, 10738,  3578,  511;
  9: 1, 1031, 8212, 28700, 57358, 71666, 57316, 28652, 8185, 1023;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*P[n - 2, x]];
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] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (2 + 4*x + 2*x^2)*v[i - 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 (−∞, 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 11

3 more

Related ?

Same fingerprint ?

A308737 → reverse

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