A105868: Triangle read by rows, T(n,k) = C(n,k)*C(k,n-k).

A105868 ↗ nonnegative coefficients

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

Recurrence from A105868

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

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

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

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	2	1													3
3	0	0	6	1												7
4	0	0	6	12	1											19
5	0	0	0	30	20	1										51
6	0	0	0	20	90	30	1									141
7	0	0	0	0	140	210	42	1								393
8	0	0	0	0	70	560	420	56	1							1107
9	0	0	0	0	0	630	1680	756	72	1						3139
10	0	0	0	0	0	252	3150	4200	1260	90	1					8953
11	0	0	0	0	0	0	2772	11550	9240	1980	110	1				25653
12	0	0	0	0	0	0	924	16632	34650	18480	2970	132	1			73789
13	0	0	0	0	0	0	0	12012	72072	90090	34320	4290	156	1		212941
14	0	0	0	0	0	0	0	3432	84084	252252	210210	60060	6006	182	1	616227

b-files: A002426 (row sums), A089627 (reversed triangle), A105868, A391403 (reversed triangle) ?

Technical details

Node id: n-6fa6977267f2e73d · revision 0 · JSON
Definition source: oeis_catalog:A105868
Fingerprint: f6-d272e4b2958d4a94

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 ?

A391403 reversed triangle 56 terms, 14 rows · tabf · nonn · catalog b-file?
A105868 triangle 80 terms, 12 rows · tabl · nonn · catalog b-file?
A089627 reversed triangle 120 terms, 15 rows · tabl · nonn · catalog b-file?

Row sums ? A002426

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 2, 1;
  3: 0, 0, 6,  1;
  4: 0, 0, 6, 12,   1;
  5: 0, 0, 0, 30,  20,   1;
  6: 0, 0, 0, 20,  90,  30,    1;
  7: 0, 0, 0,  0, 140, 210,   42,   1;
  8: 0, 0, 0,  0,  70, 560,  420,  56,  1;
  9: 0, 0, 0,  0,   0, 630, 1680, 756, 72, 1;

%F Formula

The row polynomials satisfy n*P(n,x) = (2*n-1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*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[Together[((2*n - 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*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

A089627 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  0;
  2: 1,  2,   0;
  3: 1,  6,   0,    0;
  4: 1, 12,   6,    0,   0;
  5: 1, 20,  30,    0,   0, 0;
  6: 1, 30,  90,   20,   0, 0, 0;
  7: 1, 42, 210,  140,   0, 0, 0, 0;
  8: 1, 56, 420,  560,  70, 0, 0, 0, 0;
  9: 1, 72, 756, 1680, 630, 0, 0, 0, 0, 0;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where n*P(n,x) = (2*n-1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*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[Together[((2*n - 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*P[n - 2, x])/(n)]];
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*n - 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*v[i - 2])/(n)); 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

A105868 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 2, 1;
  3: 0, 0, 6,  1;
  4: 0, 0, 6, 12,   1;
  5: 0, 0, 0, 30,  20,   1;
  6: 0, 0, 0, 20,  90,  30,    1;
  7: 0, 0, 0,  0, 140, 210,   42,   1;
  8: 0, 0, 0,  0,  70, 560,  420,  56,  1;
  9: 0, 0, 0,  0,   0, 630, 1680, 756, 72, 1;

%F Formula

The row polynomials satisfy n*P(n,x) = (2*n-1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*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[Together[((2*n - 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*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

A391403 reversed triangle, offset 0, from P₀, low zeros dropped

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 1,  2;
  3: 1,  6;
  4: 1, 12,   6;
  5: 1, 20,  30;
  6: 1, 30,  90,   20;
  7: 1, 42, 210,  140;
  8: 1, 56, 420,  560,  70;
  9: 1, 72, 756, 1680, 630;

%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) = (2*n-1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*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[Together[((2*n - 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*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

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