A111006 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 6\,x + 4\,x^{2}\right)\,P_{n-2}(x) + \left(1 + 2\,x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	3	10	8													21
3	5	26	44	24												99
4	11	74	184	200	80											549
5	21	178	596	984	800	256										2835
6	43	434	1808	3976	4864	3136	832									15093
7	85	1002	5020	13848	22704	22112	11840	2688								79299
8	171	2298	13416	44424	91216	118880	96000	43904	8704							419013
9	341	5154	34404	133080	328640	537152	580928	400768	160000	28160						2208627
10	683	11458	86016	380424	1097440	2157184	2925376	2701952	1626368	576000	91136					11654037
11	1365	25178	210028	1045656	3451664	7930528	12939072	14988416	12078592	6448128	2052096	294912				61465635
12	2731	54922	503896	2788488	10364464	27254560	51983744	72451456	73221376	52323328	25092096	7249920	954368			324245349
13	5461	118930	1190324	7248472	29958816	88737152	193760832	315752320	383968256	344087552	220846080	96129024	25432064	3088384		1710323667
14	10923	256082	2776432	18449160	83931200	276504640	680187840	1269097600	1804559360	1945804800	1566030848	912166912	363466752	88678400	9994240	9021915189

Technical details

Node id: n-908d9fd8f65f4e8f · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-8922cf5368d67490

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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:   1;
  1:   1,    2;
  2:   3,   10,     8;
  3:   5,   26,    44,     24;
  4:  11,   74,   184,    200,     80;
  5:  21,  178,   596,    984,    800,    256;
  6:  43,  434,  1808,   3976,   4864,   3136,    832;
  7:  85, 1002,  5020,  13848,  22704,  22112,  11840,   2688;
  8: 171, 2298, 13416,  44424,  91216, 118880,  96000,  43904,   8704;
  9: 341, 5154, 34404, 133080, 328640, 537152, 580928, 400768, 160000, 28160;

%F Formula

The row polynomials satisfy P(n,x) = (2+6*x+4*x^2)*P(n-2,x) + (1+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 6*x + 4*x^2)*P[n - 2, x] + (1 + 2*x)*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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 6*x + 4*x^2)*v[i - 2] + (1 + 2*x)*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 Another version of Fibonacci-Pascal triangle A037027.?
  - 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