A081582 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(7 + 7\,x\right)\,P_{n-2}(x) + \left(2 + x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	11	11	1						23
3	36	54	20	1					111
4	149	298	178	29	1				655
5	550	1375	1172	383	38	1			3519
6	2143	6429	7051	3387	669	47	1		19727
7	8136	28476	38360	24710	7672	1036	56	1	108447

Show all 15 stored rows

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(17 + 2\,x - 23\,n - n x + 8\,n^{2}\right)\,P_{n-1}(x) + \left(80 + 83\,x + 4\,x^{2} - 32\,n - 32\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(32 + 64\,x + 33\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)

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

Technical details

Node id: n-9f4396728047edfb · revision 2 · JSON
Definition source: derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-9a46395478446a6a)
Fingerprint: f6-c6734442abd299d3

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      1;
  1:      2,      1;
  2:     11,     11,      1;
  3:     36,     54,     20,      1;
  4:    149,    298,    178,     29,      1;
  5:    550,   1375,   1172,    383,     38,      1;
  6:   2143,   6429,   7051,   3387,    669,     47,     1;
  7:   8136,  28476,  38360,  24710,   7672,   1036,    56,    1;
  8:  31273, 125092, 199556, 160846,  68446,  14756,  1484,   65,  1;
  9: 119498, 537741, 992056, 962738, 524412, 158914, 25368, 2013, 74, 1;

%F Formula

The row polynomials satisfy P(n,x) = (7+7*x)*P(n-2,x) + (2+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(7 + 7*x)*P[n - 2, x] + (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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (7 + 7*x)*v[i - 2] + (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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-c5fd3da4c5a13f73
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-03aefd7a03322503, e-c5fd3da4c5a13f73
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-03aefd7a03322503, e-c5fd3da4c5a13f73
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-03aefd7a03322503, e-c5fd3da4c5a13f73
Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-03aefd7a03322503

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A007318 e-c5fd3da4c5a13f73

Geometric basis map (elementary) polytool ?
- premise (sequence): Pascal-(1,7,1) array.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-a31f5c33917b15a7 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
  - preserver_image reversal, shift 0 e-7d90ede2da8763e3
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Pascal-(1,7,1) array.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang_derivative_lag e-a31f5c33917b15a7 (expanded above)
definition_equality e-9a46395478446a6a root: theorem check

Ore-algebra identity of two definitions polytool ?
preserver_image affine, shift 0 e-03aefd7a03322503

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Pascal-(1,7,1) array.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - ↵ theorem_check liu_wang_derivative_lag e-a31f5c33917b15a7 (expanded above)
  - ↵ preserver_image reversal, shift 0 e-7d90ede2da8763e3 (expanded above)

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A081582 → x ↦ x/(x+1)

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