A128908: Riordan array (1, x/(1-x)^2).

A128908 ↗ nonnegative coefficients

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

Recurrence from A128908

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

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

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

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

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	3	4	1												8
4	0	4	10	6	1											21
5	0	5	20	21	8	1										55
6	0	6	35	56	36	10	1									144
7	0	7	56	126	120	55	12	1								377
8	0	8	84	252	330	220	78	14	1							987
9	0	9	120	462	792	715	364	105	16	1						2584
10	0	10	165	792	1716	2002	1365	560	136	18	1					6765
11	0	11	220	1287	3432	5005	4368	2380	816	171	20	1				17711
12	0	12	286	2002	6435	11440	12376	8568	3876	1140	210	22	1			46368
13	0	13	364	3003	11440	24310	31824	27132	15504	5985	1540	253	24	1		121393
14	0	14	455	4368	19448	48620	75582	77520	54264	26334	8855	2024	300	26	1	317811

b-files: A078812, A088305 (row sums), A128908, A172431 (reversed triangle) ?

Technical details

Node id: n-83d230649c15eed8 · revision 0 · JSON
Definition source: oeis_catalog:A128908
Fingerprint: f6-064b28e6ad672ed5

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 ?

A172431 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A078812 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
A128908 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A237268, A088305, A001906

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, 3,   4,   1;
  4: 0, 4,  10,   6,   1;
  5: 0, 5,  20,  21,   8,   1;
  6: 0, 6,  35,  56,  36,  10,   1;
  7: 0, 7,  56, 126, 120,  55,  12,   1;
  8: 0, 8,  84, 252, 330, 220,  78,  14,  1;
  9: 0, 9, 120, 462, 792, 715, 364, 105, 16, 1;

%F Formula

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

%t Mathematica

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

A078812 triangle, offset 0, from P₁, low zeros dropped

%e Example

Triangle begins:
  0:  1;
  1:  2,   1;
  2:  3,   4,   1;
  3:  4,  10,   6,    1;
  4:  5,  20,  21,    8,    1;
  5:  6,  35,  56,   36,   10,    1;
  6:  7,  56, 126,  120,   55,   12,   1;
  7:  8,  84, 252,  330,  220,   78,  14,   1;
  8:  9, 120, 462,  792,  715,  364, 105,  16,  1;
  9: 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1;

%F Formula

Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) - (n-1)*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = 2*x + x^2.

%t Mathematica

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

A128908 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 2,   1;
  3: 0, 3,   4,   1;
  4: 0, 4,  10,   6,   1;
  5: 0, 5,  20,  21,   8,   1;
  6: 0, 6,  35,  56,  36,  10,   1;
  7: 0, 7,  56, 126, 120,  55,  12,   1;
  8: 0, 8,  84, 252, 330, 220,  78,  14,  1;
  9: 0, 9, 120, 462, 792, 715, 364, 105, 16, 1;

%F Formula

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

%t Mathematica

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

A172431 reversed triangle, offset 1, from P₁, low zeros dropped

%e Example

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  4,   3;
   4: 1,  6,  10,   4;
   5: 1,  8,  21,  20,    5;
   6: 1, 10,  36,  56,   35,    6;
   7: 1, 12,  55, 120,  126,   56,    7;
   8: 1, 14,  78, 220,  330,  252,   84,   8;
   9: 1, 16, 105, 364,  715,  792,  462, 120,   9;
  10: 1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where (n-2)*P(n,x) = (2*n-4+(n-2)*x)*P(n-1,x) - (n-2)*P(n-2,x) for n >= 3, with P(1,x) = x, P(2,x) = 2*x + x^2.

%t Mathematica

P[1, x_] := x;
P[2, x_] := 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (n - 2)*x)*P[n - 1, x] - (n - 2)*P[n - 2, x])/(n - 2)]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = x; v[2] = 2*x + x^2; for(n = 3, N, my(i = n); v[i] = ((2*n - 4 + (n - 2)*x)*v[i - 1] - (n - 2)*v[i - 2])/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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 ≥ 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

Used in 6

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