A227450 → 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

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

\(P_{0}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	6	4							10
2	22	28	8						58
3	94	172	96	16					378
4	454	1068	872	288	32				2714
5	2430	6924	7360	3600	800	64			21178
6	14214	47276	61448	39520	13120	2112	128		177818
7	89918	340428	520352	412496	181440	43904	5376	256	1594170

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-1bf471426fab0fa9 · revision 1 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-6319fdf993d134a2

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:       2;
  1:       6,        4;
  2:      22,       28,        8;
  3:      94,      172,       96,       16;
  4:     454,     1068,      872,      288,       32;
  5:    2430,     6924,     7360,     3600,      800,       64;
  6:   14214,    47276,    61448,    39520,    13120,     2112,     128;
  7:   89918,   340428,   520352,   412496,   181440,    43904,    5376,    256;
  8:  610182,  2582252,  4520104,  4241440,  2314592,   746368,  137984,  13312,   512;
  9: 4412798, 20589580, 40489344, 43747216, 28416864, 11428032, 2827776, 413184, 32256, 1024;

%F Formula

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

%t Mathematica

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 1] + (1 + x)*deriv(v[i - 1], x)); 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, -1]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-c012cb3d271aa10d
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-c012cb3d271aa10d
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-c012cb3d271aa10d
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-c012cb3d271aa10d

Proof chains

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

preserver_image affine, shift 0 e-c012cb3d271aa10d

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-74361f30d2bc482d root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

No candidate equalities or related sequences.

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