A111006 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(14 + 17\,x + 5\,x^{2} - 13\,n - 16\,n x - 5\,n x^{2}\right)\,P_{n-1}(x) + \left(18 + 37\,x + 24\,x^{2} + 5\,x^{3}\right)\,P_{n-1}'(x)\)

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

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	3	5	2													10
3	5	13	11	3												32
4	11	37	46	25	5											124
5	21	89	149	123	50	8										440
6	43	217	452	497	304	98	13									1624
7	85	501	1255	1731	1419	691	185	21								5888
8	171	1149	3354	5553	5701	3715	1500	343	34							21520
9	341	2577	8601	16635	20540	16786	9077	3131	625	55						78368
10	683	5729	21504	47553	68590	67412	45709	21109	6353	1125	89					285856
11	1365	12589	52507	130707	215729	247829	202173	117097	47182	12594	2004	144				1041920
12	2731	27461	125974	348561	647779	851705	812246	566027	286021	102194	24504	3540	233			3798976
13	5461	59465	297581	906059	1872426	2773036	3027513	2466815	1499876	672046	215670	46938	6209	377		13849472
14	10923	128041	694108	2306145	5245700	8640770	10627935	9914825	7049060	3800400	1529327	445394	88737	10825	610	50492800

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(2 + 3\,x + x^{2}\right)\,P_{n-2}(x) + \left(1 + x\right)\,P_{n-1}(x)\)

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

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

Technical details

Node id: n-233801744e594f7f · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-5c7ea286eda729be)
Fingerprint: f6-d9ce7b485fab7bd3

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,    1;
  2:   3,    5,    2;
  3:   5,   13,   11,     3;
  4:  11,   37,   46,    25,     5;
  5:  21,   89,  149,   123,    50,     8;
  6:  43,  217,  452,   497,   304,    98,   13;
  7:  85,  501, 1255,  1731,  1419,   691,  185,   21;
  8: 171, 1149, 3354,  5553,  5701,  3715, 1500,  343,  34;
  9: 341, 2577, 8601, 16635, 20540, 16786, 9077, 3131, 625, 55;

%F Formula

The row polynomials satisfy n*P(n,x) = -(13*n-14+(16*n-17)*x+(5*n-5)*x^2)*P(n-1,x) + (18+37*x+24*x^2+5*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(13*n - 14 + (16*n - 17)*x + (5*n - 5)*x^2)*P[n - 1, x] + (18 + 37*x + 24*x^2 + 5*x^3)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(13*n - 14 + (16*n - 17)*x + (5*n - 5)*x^2)*v[i - 1] + (18 + 37*x + 24*x^2 + 5*x^3)*deriv(v[i - 1], x))/(n)); 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by 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
Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Another version of Fibonacci-Pascal triangle A037027.?
  - ↑ Liu–Wang theorem for three-term recurrences (above)

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