A105793: Expansion of e.g.f. (1 + y)^(1 + x).

A028338 → x ↦ −2x−3

A105793 only rational roots

Proved for all n: all roots real, in [−3/2, ∞) · interlacing (weak) what this means

Recurrence from OEIS A105793

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	0	1	1						2
3	0	-1	0	1					0
4	0	2	-1	-2	1				0
5	0	-6	5	5	-5	1			0
6	0	24	-26	-15	25	-9	1		0
7	0	-120	154	49	-140	70	-14	1	0

Show all 15 stored rows

Generated b-files: A105793 ?

Computed in your browser ?

Technical details

Node id: n-ebe1ce47fcd8575c · revision 1 · JSON
Definition source: oeis_catalog:A105793
Fingerprint: f6-f072b8c7a5a24af8

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): no
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

A180215: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A105793: triangle, 66 terms agree, 11 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 1,     1;
  2: 0,     1,     1;
  3: 0,    -1,     0,    1;
  4: 0,     2,    -1,   -2,     1;
  5: 0,    -6,     5,    5,    -5,    1;
  6: 0,    24,   -26,  -15,    25,   -9,     1;
  7: 0,  -120,   154,   49,  -140,   70,   -14,   1;
  8: 0,   720, -1044, -140,   889, -560,   154, -20,   1;
  9: 0, -5040,  8028,  -64, -6363, 4809, -1638, 294, -27, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(n-2-x)*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[-(n - 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; for(n = 1, N, my(i = n + 1); v[i] = -(n - 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.

A105793: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,     1;
  2: 0,     1,     1;
  3: 0,    -1,     0,    1;
  4: 0,     2,    -1,   -2,     1;
  5: 0,    -6,     5,    5,    -5,    1;
  6: 0,    24,   -26,  -15,    25,   -9,     1;
  7: 0,  -120,   154,   49,  -140,   70,   -14,   1;
  8: 0,   720, -1044, -140,   889, -560,   154, -20,   1;
  9: 0, -5040,  8028,  -64, -6363, 4809, -1638, 294, -27, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(n-2-x)*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[-(n - 2 - x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = -(n - 2 - x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 [-3/2, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-6cd09736036c3bec
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-6cd09736036c3bec

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-6cd09736036c3bec

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-59b849691f316c6f root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers root_domain e-ef36674c01230a86 root: theorem check
    
    Parent-row provers of polytool polytool ?

Used in 2 proof steps

e-167ee17f7f3ebd58 obreschkoff_combination → obreschkoff_combination of Expansion of e.g.f. (1 + y)^(1 + x).
e-167ee17f7f3ebd58 obreschkoff_combination → obreschkoff_combination of Expansion of e.g.f. (1 + y)^(1 + x).

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