A121314: Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

A121314 ↗ nonnegative coefficients

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

Recurrence from A121314

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

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

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

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	1													2
3	0	1	3	1												5
4	0	1	5	6	1											13
5	0	1	7	15	10	1										34
6	0	1	9	28	35	15	1									89
7	0	1	11	45	84	70	21	1								233
8	0	1	13	66	165	210	126	28	1							610
9	0	1	15	91	286	495	462	210	36	1						1597
10	0	1	17	120	455	1001	1287	924	330	45	1					4181
11	0	1	19	153	680	1820	3003	3003	1716	495	55	1				10946
12	0	1	21	190	969	3060	6188	8008	6435	3003	715	66	1			28657
13	0	1	23	231	1330	4845	11628	18564	19448	12870	5005	1001	78	1		75025
14	0	1	25	276	1771	7315	20349	38760	50388	43758	24310	8008	1365	91	1	196418

b-files: A001519 (row sums), A048575 (row sums), A054142, A085478 (reversed triangle), A121314, A122367 (row sums), A165253 (reversed triangle) ?

Technical details

Node id: n-3af57aaf5c129e4b · revision 0 · JSON
Definition source: oeis_catalog:A121314
Fingerprint: f6-2f8272d81c3bae1a

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 ?

A165253 reversed triangle 74 terms, 11 rows · tabl · nonn · catalog b-file?
A085478 reversed triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
A054142 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
A121314 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A048575, A001519, A122367

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1,  1;
  3: 0, 1,  3,  1;
  4: 0, 1,  5,  6,   1;
  5: 0, 1,  7, 15,  10,   1;
  6: 0, 1,  9, 28,  35,  15,   1;
  7: 0, 1, 11, 45,  84,  70,  21,   1;
  8: 0, 1, 13, 66, 165, 210, 126,  28,  1;
  9: 0, 1, 15, 91, 286, 495, 462, 210, 36, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,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[(1 + 2*x)*P[n - 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] = x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] - 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

A054142 triangle, offset 0, from P₁, low zeros dropped

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,   1;
  3: 1,  5,   6,   1;
  4: 1,  7,  15,  10,    1;
  5: 1,  9,  28,  35,   15,    1;
  6: 1, 11,  45,  84,   70,   21,   1;
  7: 1, 13,  66, 165,  210,  126,  28,   1;
  8: 1, 15,  91, 286,  495,  462, 210,  36,  1;
  9: 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 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) = (1+2*x)*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = x + x^2.

%t Mathematica

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

%o PARI/GP

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

A085478 reversed triangle, offset 0, from P₁, low zeros dropped

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,   1;
  3: 1,  6,   5,   1;
  4: 1, 10,  15,   7,    1;
  5: 1, 15,  35,  28,    9,    1;
  6: 1, 21,  70,  84,   45,   11,   1;
  7: 1, 28, 126, 210,  165,   66,  13,   1;
  8: 1, 36, 210, 462,  495,  286,  91,  15,  1;
  9: 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1;

%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) = (1+2*x)*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = x + x^2.

%t Mathematica

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

%o PARI/GP

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

A121314 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1,  1;
  3: 0, 1,  3,  1;
  4: 0, 1,  5,  6,   1;
  5: 0, 1,  7, 15,  10,   1;
  6: 0, 1,  9, 28,  35,  15,   1;
  7: 0, 1, 11, 45,  84,  70,  21,   1;
  8: 0, 1, 13, 66, 165, 210, 126,  28,  1;
  9: 0, 1, 15, 91, 286, 495, 462, 210, 36, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,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[(1 + 2*x)*P[n - 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] = x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] - 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

A165253 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  0;
  2: 1,  1,   0;
  3: 1,  3,   1,   0;
  4: 1,  6,   5,   1,   0;
  5: 1, 10,  15,   7,   1,   0;
  6: 1, 15,  35,  28,   9,   1,  0;
  7: 1, 21,  70,  84,  45,  11,  1,  0;
  8: 1, 28, 126, 210, 165,  66, 13,  1, 0;
  9: 1, 36, 210, 462, 495, 286, 91, 15, 1, 0;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+2*x)*P(n-1,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[(1 + 2*x)*P[n - 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] = x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] - 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

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