A236471: Riordan array ((1-x)/(1-2x), x(1-x)/(1-2x)^2).

A236471 ↗ nonnegative coefficients

Proved for all n all roots real, in [−4, ∞)interlacing (weak)

Recurrence from A236471

\(\left(-2 + n\right)\,P_{n}(x) = \left(-8 - 2\,x + 4\,n + n x\right)\,P_{n-1}(x) + \left(8 + 2\,x - 4\,n - n x\right)\,P_{n-2}(x)\)

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	2	4	1													7
3	4	13	7	1												25
4	8	38	33	10	1											90
5	16	104	129	62	13	1										325
6	32	272	450	304	100	16	1									1175
7	64	688	1452	1289	590	147	19	1								4250
8	128	1696	4424	4942	2945	1014	203	22	1							15375
9	256	4096	12896	17584	13073	5823	1603	268	25	1						55625
10	512	9728	36288	59040	53154	29364	10409	2384	342	28	1					201250
11	1024	22784	99200	189216	201780	134245	58765	17270	3384	425	31	1				728125
12	2048	52736	264704	583616	724680	568150	298305	107900	27054	4630	517	34	1			2634375
13	4096	120832	691968	1743104	2486000	2258520	1392065	602060	185310	40490	6149	618	37	1		9531250
14	8192	274432	1777152	5065216	8204768	8522800	6065410	3070400	1127184	301696	58388	7968	728	40	1	34484375

b-files: A052936 (row sums), A236471 ?

Technical details

Node id: n-44248dd6011d9846 · revision 0 · JSON
Definition source: oeis_catalog:A236471
Fingerprint: f6-39dc65612e9c69b1

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 ?

A236471 triangle 57 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A052936

Submission text ?

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

%e Example

Triangle begins:
  0:   1;
  1:   1,    1;
  2:   2,    4,     1;
  3:   4,   13,     7,     1;
  4:   8,   38,    33,    10,     1;
  5:  16,  104,   129,    62,    13,    1;
  6:  32,  272,   450,   304,   100,   16,    1;
  7:  64,  688,  1452,  1289,   590,  147,   19,   1;
  8: 128, 1696,  4424,  4942,  2945, 1014,  203,  22,  1;
  9: 256, 4096, 12896, 17584, 13073, 5823, 1603, 268, 25, 1;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(n-2)*x)*P(n-1,x) - (4*n-8+(n-2)*x)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 2 + 4*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 2 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (n - 2)*x)*P[n - 1, x] - (4*n - 8 + (n - 2)*x)*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 + x; v[3] = 2 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8 + (n - 2)*x)*v[i - 1] - (4*n - 8 + (n - 2)*x)*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

A236471 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:   1;
  1:   1,    1;
  2:   2,    4,     1;
  3:   4,   13,     7,     1;
  4:   8,   38,    33,    10,     1;
  5:  16,  104,   129,    62,    13,    1;
  6:  32,  272,   450,   304,   100,   16,    1;
  7:  64,  688,  1452,  1289,   590,  147,   19,   1;
  8: 128, 1696,  4424,  4942,  2945, 1014,  203,  22,  1;
  9: 256, 4096, 12896, 17584, 13073, 5823, 1603, 268, 25, 1;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(n-2)*x)*P(n-1,x) - (4*n-8+(n-2)*x)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 2 + 4*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 2 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (n - 2)*x)*P[n - 1, x] - (4*n - 8 + (n - 2)*x)*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 + x; v[3] = 2 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8 + (n - 2)*x)*v[i - 1] - (4*n - 8 + (n - 2)*x)*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 [-4, +∞)
by 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

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