A098158: Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).

A098158 ↗ nonnegative coefficients

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

Recurrence from A098158

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	1														1
3	0	1	1													2
4	0	0	3	1												4
5	0	0	1	6	1											8
6	0	0	0	5	10	1										16
7	0	0	0	1	15	15	1									32
8	0	0	0	0	7	35	21	1								64
9	0	0	0	0	1	28	70	28	1							128
10	0	0	0	0	0	9	84	126	36	1						256
11	0	0	0	0	0	1	45	210	210	45	1					512
12	0	0	0	0	0	0	11	165	462	330	55	1				1024
13	0	0	0	0	0	0	1	66	495	924	495	66	1			2048
14	0	0	0	0	0	0	0	13	286	1287	1716	715	78	1		4096
15	0	0	0	0	0	0	0	1	91	1001	3003	3003	1001	91	1	8192

b-files: A011782 (row sums), A034839 (reversed triangle), A098158, A109446, A202023 (reversed triangle) ?

Technical details

Node id: n-cf23198864b5678a · revision 0 · JSON
Definition source: oeis_catalog:A098158
Fingerprint: f6-4e15cddc85973b47

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 -1)
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 ?

A098158 triangle 88 terms, 12 rows · tabl · nonn · catalog b-file?
A109446 triangle 64 terms, 15 rows · tabf · nonn · catalog b-file?
A034839 reversed triangle 56 terms, 14 rows · tabf · nonn · catalog b-file?
A202023 reversed triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A249169, A274866, A274864, A274865, A220469, A220493, A011782, A166444

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 0, 1;
   3: 0, 1, 1;
   4: 0, 0, 3, 1;
   5: 0, 0, 1, 6,  1;
   6: 0, 0, 0, 5, 10,  1;
   7: 0, 0, 0, 1, 15, 15,  1;
   8: 0, 0, 0, 0,  7, 35, 21,   1;
   9: 0, 0, 0, 0,  1, 28, 70,  28,  1;
  10: 0, 0, 0, 0,  0,  9, 84, 126, 36, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] + (x - x^2)*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; v[2] = x; for(n = 3, N, my(i = n); v[i] = 2*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, v[i])))};
T(10)

polytool c48629374597

A034839 reversed triangle, offset 0, from P₁, low zeros dropped

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 1,  1;
  3: 1,  3;
  4: 1,  6,   1;
  5: 1, 10,   5;
  6: 1, 15,  15,  1;
  7: 1, 21,  35,  7;
  8: 1, 28,  70, 28, 1;
  9: 1, 36, 126, 84, 9;

%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) = 2*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[2*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] = 2*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

A098158 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1, 1;
  3: 0, 0, 3, 1;
  4: 0, 0, 1, 6,  1;
  5: 0, 0, 0, 5, 10,  1;
  6: 0, 0, 0, 1, 15, 15,  1;
  7: 0, 0, 0, 0,  7, 35, 21,   1;
  8: 0, 0, 0, 0,  1, 28, 70,  28,  1;
  9: 0, 0, 0, 0,  0,  9, 84, 126, 36, 1;

%F Formula

The row polynomials satisfy P(n,x) = 2*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[2*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] = 2*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

A109446 triangle, offset 0, from P₁, low zeros dropped

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 1,  1;
  3: 3,  1;
  4: 1,  6,   1;
  5: 5, 10,   1;
  6: 1, 15,  15,  1;
  7: 7, 35,  21,  1;
  8: 1, 28,  70, 28, 1;
  9: 9, 84, 126, 36, 1;

%F Formula

Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = 2*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[2*x*P[n - 1, x] + (x - x^2)*P[n - 2, x]];
Table[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] = 2*x*v[i - 1] + (x - x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A202023 reversed triangle, offset 0, from P₁

%e Example

Triangle begins:
  0: 1;
  1: 1,  0;
  2: 1,  1,   0;
  3: 1,  3,   0,  0;
  4: 1,  6,   1,  0, 0;
  5: 1, 10,   5,  0, 0, 0;
  6: 1, 15,  15,  1, 0, 0, 0;
  7: 1, 21,  35,  7, 0, 0, 0, 0;
  8: 1, 28,  70, 28, 1, 0, 0, 0, 0;
  9: 1, 36, 126, 84, 9, 0, 0, 0, 0, 0;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = 2*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[2*x*P[n - 1, x] + (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] = x; for(n = 2, N, my(i = n + 1); v[i] = 2*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 = 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 ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from 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

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