A201701: Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

A201701 ↗ nonnegative coefficients

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

Recurrence from A201701

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	2	1							3
3	4	3							7
4	8	8	1						17
5	16	20	5						41
6	32	48	18	1					99
7	64	112	56	7					239
8	128	256	160	32	1				577
9	256	576	432	120	9				1393
10	512	1280	1120	400	50	1			3363
11	1024	2816	2816	1232	220	11			8119
12	2048	6144	6912	3584	840	72	1		19601
13	4096	13312	16640	9984	2912	364	13		47321
14	8192	28672	39424	26880	9408	1568	98	1	114243

b-files: A001333 (row sums), A078057 (row sums), A207537 ?

Technical details

Node id: n-495335f6a4ca3289 · revision 0 · JSON
Definition source: oeis_catalog:A201701
Fingerprint: f6-702b8764f2c773f5

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 ?

A207537 triangle 55 terms, 13 rows · tabf · nonn b-file?

Row sums ? A078057, A001333

Submission text ?

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

%e Example

Triangle begins:
  0:   1;
  1:   1;
  2:   2,   1;
  3:   4,   3;
  4:   8,   8,   1;
  5:  16,  20,   5;
  6:  32,  48,  18,   1;
  7:  64, 112,  56,   7;
  8: 128, 256, 160,  32, 1;
  9: 256, 576, 432, 120, 9;

%F Formula

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

%t Mathematica

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

A207537 triangle, offset 1, from P₁

%e Example

Triangle begins:
   1:   1;
   2:   2,    1;
   3:   4,    3;
   4:   8,    8,    1;
   5:  16,   20,    5;
   6:  32,   48,   18,   1;
   7:  64,  112,   56,   7;
   8: 128,  256,  160,  32,  1;
   9: 256,  576,  432, 120,  9;
  10: 512, 1280, 1120, 400, 50, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] + x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = 2*v[i - 1] + x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); 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 ≥ 1
by Liu–Wang theorem for three-term recurrences
Basis map preserves real-rootedness? Basis map u^k -> P_k sends roots in (−∞, 0] to roots in (−∞, 0]: every PF input has a real-rooted image with every root <= -c/b = 0
by Constant-current basis map
Basis map preserves real-rootedness? Basis map u^k -> P_k sends roots in (−∞, 0] to roots in (−∞, 0]: every PF input has a PF image (zero allowed)
by Constant-current basis map

Proof chains ? JSON

Constant-current basis map [Bra15, Lemma 8.5 of arXiv:1410.6601] polytool RealRooted orderTwoBasis_pf_equivalences RealRooted basisTransform_orderTwoBasis_preserves_bounded_pf_iff ? 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

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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