A084534: Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).

A084534 ↗ nonnegative coefficients

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

Recurrence from A084534

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 - 4\,x + n + 2\,n x\right)\,P_{n-1}(x) + \left(2\,x^{2} - n x^{2}\right)\,P_{n-2}(x)\)

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

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

\(P_{2}(x) = 1 + 4\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	4	2						7
3	1	6	9	2					18
4	1	8	20	16	2				47
5	1	10	35	50	25	2			123
6	1	12	54	112	105	36	2		322
7	1	14	77	210	294	196	49	2	843

All 15 rows b-files: A084534 ?

Technical details

Node id: n-2496e2f555756362 · revision 0 · JSON
Definition source: oeis_catalog:A084534
Fingerprint: f6-4a6fdd8c1c3f626a

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 ?

A084534 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A005248

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  4,   2;
  3: 1,  6,   9,   2;
  4: 1,  8,  20,  16,    2;
  5: 1, 10,  35,  50,   25,    2;
  6: 1, 12,  54, 112,  105,   36,    2;
  7: 1, 14,  77, 210,  294,  196,   49,   2;
  8: 1, 16, 104, 352,  660,  672,  336,  64,  2;
  9: 1, 18, 135, 546, 1287, 1782, 1386, 540, 81, 2;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(2*n-4)*x)*P(n-1,x) - (n-2)*x^2*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 4*x + 2*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 4*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (2*n - 4)*x)*P[n - 1, x] - (n - 2)*x^2*P[n - 2, x])/(n - 2)]];
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; v[3] = 1 + 4*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (2*n - 4)*x)*v[i - 1] - (n - 2)*x^2*v[i - 2])/(n - 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

A084534 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  4,   2;
  3: 1,  6,   9,   2;
  4: 1,  8,  20,  16,    2;
  5: 1, 10,  35,  50,   25,    2;
  6: 1, 12,  54, 112,  105,   36,    2;
  7: 1, 14,  77, 210,  294,  196,   49,   2;
  8: 1, 16, 104, 352,  660,  672,  336,  64,  2;
  9: 1, 18, 135, 546, 1287, 1782, 1386, 540, 81, 2;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(2*n-4)*x)*P(n-1,x) - (n-2)*x^2*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 4*x + 2*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 4*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (2*n - 4)*x)*P[n - 1, x] - (n - 2)*x^2*P[n - 2, x])/(n - 2)]];
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; v[3] = 1 + 4*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (2*n - 4)*x)*v[i - 1] - (n - 2)*x^2*v[i - 2])/(n - 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 (−∞, +∞)
by Three-term recurrence criterion
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, Three-term recurrence criterion
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, Three-term recurrence criterion
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences, Three-term recurrence criterion
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m ≥ 2
by Three-term recurrence criterion
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
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
Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Used in 12

4 more

Related ?

Same fingerprint ?

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	4	2						7
3	1	6	9	2					18
4	1	8	20	16	2				47
5	1	10	35	50	25	2			123
6	1	12	54	112	105	36	2		322
7	1	14	77	210	294	196	49	2	843

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	4	2						7
3	1	6	9	2					18
4	1	8	20	16	2				47
5	1	10	35	50	25	2			123
6	1	12	54	112	105	36	2		322
7	1	14	77	210	294	196	49	2	843

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	4	2						7
3	1	6	9	2					18
4	1	8	20	16	2				47
5	1	10	35	50	25	2			123
6	1	12	54	112	105	36	2		322
7	1	14	77	210	294	196	49	2	843