A111006: Another version of Fibonacci-Pascal triangle A037027.

A111006 ↗ nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence from A111006

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	1														1
2	0	1	2													3
3	0	0	2	3												5
4	0	0	1	5	5											11
5	0	0	0	3	10	8										21
6	0	0	0	1	9	20	13									43
7	0	0	0	0	4	22	38	21								85
8	0	0	0	0	1	14	51	71	34							171
9	0	0	0	0	0	5	40	111	130	55						341
10	0	0	0	0	0	1	20	105	233	235	89					683
11	0	0	0	0	0	0	6	65	256	474	420	144				1365
12	0	0	0	0	0	0	1	27	190	594	942	744	233			2731
13	0	0	0	0	0	0	0	7	98	511	1324	1836	1308	377		5461
14	0	0	0	0	0	0	0	1	35	315	1295	2860	3522	2285	610	10923

b-files: A111006, A128100 (reversed triangle), A283642 (row sums) ?

Technical details

Node id: n-e7c079f81f452bf3 · revision 0 · JSON
Definition source: oeis_catalog:A111006
Fingerprint: f6-c12a5875baa63cc4

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 ?

A128100 reversed triangle 64 terms, 15 rows · tabf · nonn · catalog b-file?
A111006 triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?

Row sums ? A152046, A001045, A168642, A167167, A283642

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1, 2;
  3: 0, 0, 2, 3;
  4: 0, 0, 1, 5,  5;
  5: 0, 0, 0, 3, 10,  8;
  6: 0, 0, 0, 1,  9, 20, 13;
  7: 0, 0, 0, 0,  4, 22, 38,  21;
  8: 0, 0, 0, 0,  1, 14, 51,  71,  34;
  9: 0, 0, 0, 0,  0,  5, 40, 111, 130, 55;

%F Formula

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

%t Mathematica

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

A111006 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1, 2;
  3: 0, 0, 2, 3;
  4: 0, 0, 1, 5,  5;
  5: 0, 0, 0, 3, 10,  8;
  6: 0, 0, 0, 1,  9, 20, 13;
  7: 0, 0, 0, 0,  4, 22, 38,  21;
  8: 0, 0, 0, 0,  1, 14, 51,  71,  34;
  9: 0, 0, 0, 0,  0,  5, 40, 111, 130, 55;

%F Formula

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

%t Mathematica

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

A128100 reversed triangle, offset 0, from P₀, low zeros dropped

%e Example

Triangle begins:
  0:  1;
  1:  1;
  2:  2,   1;
  3:  3,   2;
  4:  5,   5,   1;
  5:  8,  10,   3;
  6: 13,  20,   9,  1;
  7: 21,  38,  22,  4;
  8: 34,  71,  51, 14, 1;
  9: 55, 130, 111, 40, 5;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (x + x^2)*P[n - 2, x]];
Table[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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 [-1, 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 13

5 more

Related ?

Same fingerprint ?

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