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

A115951 ↗ nonnegative coefficients

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

Recurrence from A115951

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

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

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

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

b-files: A006139 (row sums), A008556 (reversed triangle), A115951 ?

Technical details

Node id: n-eacb751f39dbd056 · revision 0 · JSON
Definition source: oeis_catalog:A115951
Fingerprint: f6-a866a9bf376c404a

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 ?

A115951 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
A008556 reversed 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: 0, 2;
  2: 0, 2,  6;
  3: 0, 0, 12, 20;
  4: 0, 0,  6, 60,  70;
  5: 0, 0,  0, 60, 280,  252;
  6: 0, 0,  0, 20, 420, 1260,   924;
  7: 0, 0,  0,  0, 280, 2520,  5544,  3432;
  8: 0, 0,  0,  0,  70, 2520, 13860, 24024,  12870;
  9: 0, 0,  0,  0,   0, 1260, 18480, 72072, 102960, 48620;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 2)*x*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*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 2)*x*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 reversed triangle, offset 0, from P₀, low zeros dropped

%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

Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where n*P(n,x) = (4*n-2)*x*P(n-1,x) + (4*n-4)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.

%t Mathematica

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

A115951 triangle, offset 0, from P₀

%e Example

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

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 2)*x*P[n - 1, x] + (4*n - 4)*x*P[n - 2, x])/(n)]];
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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 2)*x*v[i - 1] + (4*n - 4)*x*v[i - 2])/(n)); 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 (−∞, 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 12

4 more

Related ?

Same fingerprint ?

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

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