A330510: Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <= n.

A330510 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A330510

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

\(P_{0}(x) = 2\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	3	3														6
2	4	10	4													18
3	5	22	22	5												54
4	6	40	70	40	6											162
5	7	65	171	171	65	7										486
6	8	98	356	534	356	98	8									1458
7	9	140	665	1373	1373	665	140	9								4374
8	10	192	1148	3088	4246	3088	1148	192	10							13122
9	11	255	1866	6294	11257	11257	6294	1866	255	11						39366
10	12	330	2892	11892	26620	34606	26620	11892	2892	330	12					118098
11	13	418	4312	21153	57607	93644	93644	57607	21153	4312	418	13				354294
12	14	520	6226	35816	116116	229384	286730	229384	116116	35816	6226	520	14			1062882
13	15	637	8749	58201	220792	518596	787333	787333	518596	220792	58201	8749	637	15		3188646
14	16	770	12012	91338	399828	1097460	1979628	2403834	1979628	1097460	399828	91338	12012	770	16	9565938

b-files: A008776 (row sums), A330510, A330510 (reversed triangle) ?

Technical details

Node id: n-539ff84504d2bf3a · revision 0 · JSON
Definition source: oeis_catalog:A330510
Fingerprint: f6-e5056e0add0ce0ee

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: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

A330510 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A330510 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A027334, A025192, A008776, A099856, A110593

Submission text ?

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

%e Example

Triangle begins:
  0:  2;
  1:  3,   3;
  2:  4,  10,    4;
  3:  5,  22,   22,    5;
  4:  6,  40,   70,   40,     6;
  5:  7,  65,  171,  171,    65,     7;
  6:  8,  98,  356,  534,   356,    98,    8;
  7:  9, 140,  665, 1373,  1373,   665,  140,    9;
  8: 10, 192, 1148, 3088,  4246,  3088, 1148,  192,  10;
  9: 11, 255, 1866, 6294, 11257, 11257, 6294, 1866, 255, 11;

%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) = 2, P(1,x) = 3 + 3*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + 3*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] = 2; v[2] = 3 + 3*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

A330510 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:  2;
  1:  3,   3;
  2:  4,  10,    4;
  3:  5,  22,   22,    5;
  4:  6,  40,   70,   40,     6;
  5:  7,  65,  171,  171,    65,     7;
  6:  8,  98,  356,  534,   356,    98,    8;
  7:  9, 140,  665, 1373,  1373,   665,  140,    9;
  8: 10, 192, 1148, 3088,  4246,  3088, 1148,  192,  10;
  9: 11, 255, 1866, 6294, 11257, 11257, 6294, 1866, 255, 11;

%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) = 2, P(1,x) = 3 + 3*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + 3*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] = 2; v[2] = 3 + 3*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

A330510 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:  2;
  1:  3,   3;
  2:  4,  10,    4;
  3:  5,  22,   22,    5;
  4:  6,  40,   70,   40,     6;
  5:  7,  65,  171,  171,    65,     7;
  6:  8,  98,  356,  534,   356,    98,    8;
  7:  9, 140,  665, 1373,  1373,   665,  140,    9;
  8: 10, 192, 1148, 3088,  4246,  3088, 1148,  192,  10;
  9: 11, 255, 1866, 6294, 11257, 11257, 6294, 1866, 255, 11;

%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) = 2, P(1,x) = 3 + 3*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + 3*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] = 2; v[2] = 3 + 3*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

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