A084950 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	10	8							18
3	43	40	4						87
4	225	228	36						489
5	1393	1494	300	8					3195
6	9976	11136	2592	128					23832
7	81201	93368	24024	1632	16				200241
8	740785	871400	241080	20000	400				1873665
9	7489051	8969770	2621560	249680	7280	32			19337373
10	83120346	101020440	30821040	3248640	120480	1152			218332098
11	1004933203	1236193152	390413580	44476480	1952400	28416	64		2677997295
12	13147251985	16337772108	5308238460	643084960	31998960	611520	3136		35468961129
13	185066460993	231974869070	77178138324	9828493080	538890800	12494496	100800	128	504599447691
14	2789144166880	3522255312128	1195655857536	158686958080	9401530880	252026880	2738176	8192	7675398598752

Technical details

Node id: n-8a9d356b74002a5f · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-5b162ab5178f30e8

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:       1;
  1:       3,       2;
  2:      10,       8;
  3:      43,      40,       4;
  4:     225,     228,      36;
  5:    1393,    1494,     300,      8;
  6:    9976,   11136,    2592,    128;
  7:   81201,   93368,   24024,   1632,   16;
  8:  740785,  871400,  241080,  20000,  400;
  9: 7489051, 8969770, 2621560, 249680, 7280, 32;

%F Formula

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

%t Mathematica

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

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+...…?
  - 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