A008556: Triangle of coefficients of Legendre polynomials 2^n P_n (x).

A008556 ↗ nonnegative coefficients

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

Recurrence from A008556

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	6	2							8
3	20	12							32
4	70	60	6						136
5	252	280	60						592
6	924	1260	420	20					2624
7	3432	5544	2520	280					11776
8	12870	24024	13860	2520	70				53344
9	48620	102960	72072	18480	1260				243392
10	184756	437580	360360	120120	13860	252			1116928
11	705432	1847560	1750320	720720	120120	5544			5149696
12	2704156	7759752	8314020	4084080	900900	72072	924		23835904
13	10400600	32449872	38798760	22170720	6126120	720720	24024		110690816
14	40116600	135207800	178474296	116396280	38798760	6126120	360360	3432	515483648

b-files: A006139 (row sums), A008556 ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

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

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

Technical details

Node id: n-b2f14d9f44a39df2 · revision 1 · JSON
Definition source: oeis_catalog:A008556
derived:reversal (equal by e-5fb73cf2df09a656)
Fingerprint: f6-a866a9bf376c404a

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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 ?

A008556 triangle 42 terms, 12 rows · tabf · nonn · catalog b-file?

Row sums ? A006139

Submission text ?

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

%e Example

Triangle begins:
  0:     1;
  1:     2;
  2:     6,      2;
  3:    20,     12;
  4:    70,     60,     6;
  5:   252,    280,    60;
  6:   924,   1260,   420,    20;
  7:  3432,   5544,  2520,   280;
  8: 12870,  24024, 13860,  2520,   70;
  9: 48620, 102960, 72072, 18480, 1260;

%F Formula

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

%t Mathematica

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

A008556 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:     1;
  1:     2;
  2:     6,      2;
  3:    20,     12;
  4:    70,     60,     6;
  5:   252,    280,    60;
  6:   924,   1260,   420,    20;
  7:  3432,   5544,  2520,   280;
  8: 12870,  24024, 13860,  2520,   70;
  9: 48620, 102960, 72072, 18480, 1260;

%F Formula

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

%t Mathematica

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

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 Affine, reversal and Möbius substitutions (elementary), Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), 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
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Expansion of 1/sqrt(1-4*x*y-4*x^2*y).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
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 10

2 more

Related ?

Same fingerprint ?

A115951: Expansion of 1/sqrt(1-4*x*y-4*x^2*y).

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