A039756 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

\(P_{1}(x) = 2 + 5\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	2	1															3
1	2	5	1														8
2	2	13	14	1													30
3	2	25	71	41	1												140
4	2	41	228	370	122	1											764
5	2	61	565	1850	1892	365	1										4736
6	2	85	1186	6595	14182	9583	1094	1									32728
7	2	113	2219	18851	70497	105315	48281	3281	1								248560
8	2	145	3816	46116	267708	714966	767048	242500	9842	1							2052144
9	2	181	6153	100500	839664	3543582	7009550	5517460	1215782	29525	1						18262400
10	2	221	9430	200325	2282484	14039970	44751212	67163690	39359562	6088753	88574	1					173984224
11	2	265	13871	371965	5551689	47017146	221421398	546847642	633276292	279193805	30473291	265721	1				1764433088
12	2	313	19724	651926	12354694	138090975	904448072	3348402772	6524829454	5902373191	1972711468	152455030	797162	1			18957134784
13	2	365	27261	1089166	25565540	365031667	3180190871	16549204100	49084141106	76485251843	54547801035	13900665870	762540872	2391485	1		214903901184
14	2	421	36778	1747655	49792106	884997113	9911414370	69169472515	291595321446	702719672655	884627094078	500946221685	97762823342	3813501523	7174454	1	2561489270144

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Technical details

Node id: n-05c4badff7ad6784 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-d3aafc179db2d446

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
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;
  1: 2,   5,    1;
  2: 2,  13,   14,      1;
  3: 2,  25,   71,     41,      1;
  4: 2,  41,  228,    370,    122,       1;
  5: 2,  61,  565,   1850,   1892,     365,       1;
  6: 2,  85, 1186,   6595,  14182,    9583,    1094,       1;
  7: 2, 113, 2219,  18851,  70497,  105315,   48281,    3281,       1;
  8: 2, 145, 3816,  46116, 267708,  714966,  767048,  242500,    9842,     1;
  9: 2, 181, 6153, 100500, 839664, 3543582, 7009550, 5517460, 1215782, 29525, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(2*n-3)*x)*P(n-2,x) - 2*x^2*d/dx P(n-2,x) + (2*n+1)*x*P(n-1,x) - 2*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 2 + 5*x + x^2.

%t Mathematica

P[0, x_] := 2 + x;
P[1, x_] := 2 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n - 3)*x)*P[n - 2, x] - 2*x^2*D[P[n - 2, x], x] + (2*n + 1)*x*P[n - 1, x] - 2*x^2*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 + x; v[2] = 2 + 5*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (2*n - 3)*x)*v[i - 2] - 2*x^2*deriv(v[i - 2], x) + (2*n + 1)*x*v[i - 1] - 2*x^2*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, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-3e597de3f3fed6c2

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

obreschkoff_combination e-3e597de3f3fed6c2

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of B-analogs of Stirling numbers of 2nd kind.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-7519e7535554cb98 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle of B-analogs of Stirling numbers of 2nd kind.
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  
  no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.

References

[Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
[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
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.