A330381: Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= n.

A330381 ↗ nonnegative coefficients

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

Recurrence from A330381

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	5	3						9
3	1	9	13	4					27
4	1	14	35	26	5				81
5	1	20	75	96	45	6			243
6	1	27	140	267	216	71	7		729
7	1	35	238	623	750	427	105	8	2187

All 15 rows b-files: A000244 (row sums), A104029 (reversed triangle), A330381 ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

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

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

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

Technical details

Node id: n-b1a7dcc8610d7e32 · revision 1 · JSON
Definition source: oeis_catalog:A330381
derived:reversal (equal by e-637632b6c2fd4634)
Fingerprint: f6-2f76d9ab212d3ec7

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 ?

A104029 reversed triangle 62 terms, 10 rows · tabl · nonn · catalog b-file?
A330381 triangle 61 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A140429, A354269, A133494, A155734, A027313, A000244, A376478

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  5,   3;
  3: 1,  9,  13,    4;
  4: 1, 14,  35,   26,    5;
  5: 1, 20,  75,   96,   45,    6;
  6: 1, 27, 140,  267,  216,   71,    7;
  7: 1, 35, 238,  623,  750,  427,  105,    8;
  8: 1, 44, 378, 1288, 2123, 1800,  770,  148,   9;
  9: 1, 54, 570, 2436, 5211, 6046, 3858, 1296, 201, 10;

%F Formula

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

%t Mathematica

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

A104029 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:  1;
  1:  2,   1;
  2:  3,   5,    1;
  3:  4,  13,    9,    1;
  4:  5,  26,   35,   14,    1;
  5:  6,  45,   96,   75,   20,    1;
  6:  7,  71,  216,  267,  140,   27,    1;
  7:  8, 105,  427,  750,  623,  238,   35,   1;
  8:  9, 148,  770, 1800, 2123, 1288,  378,  44,  1;
  9: 10, 201, 1296, 3858, 6046, 5211, 2436, 570, 54, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-1,x) - (1+x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + x + x^2)*P[n - 2, x]];
Table[Reverse[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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 + x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A330381 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  5,   3;
  3: 1,  9,  13,    4;
  4: 1, 14,  35,   26,    5;
  5: 1, 20,  75,   96,   45,    6;
  6: 1, 27, 140,  267,  216,   71,    7;
  7: 1, 35, 238,  623,  750,  427,  105,    8;
  8: 1, 44, 378, 1288, 2123, 1800,  770,  148,   9;
  9: 1, 54, 570, 2436, 5211, 6046, 3858, 1296, 201, 10;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + x + 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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 + x + 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, Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences, Affine, reversal and Möbius substitutions (elementary)
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
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= …?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
↑ Liu–Wang theorem for three-term recurrences (above)

Used in 12

4 more

Related ?

Same fingerprint ?

A104029: Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).

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