A269939 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

\(P_{2}(x) = 2\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	1	1								2
2	0	2	3							5
3	0	2	13	15						30
4	0	2	35	120	105					262
5	0	2	81	595	1365	945				2988
6	0	2	175	2408	10710	18270	10395			41960
7	0	2	365	8743	66430	207900	280665	135135		699240
8	0	2	747	29760	359975	1827210	4369365	4864860	2027025	13478944

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-ce43f3f34cdc34f8 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-4ef688dcc1f7919d

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 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1, 1;
   2: 0, 2,    3;
   3: 0, 2,   13,     15;
   4: 0, 2,   35,    120,     105;
   5: 0, 2,   81,    595,    1365,      945;
   6: 0, 2,  175,   2408,   10710,    18270,     10395;
   7: 0, 2,  365,   8743,   66430,   207900,    280665,     135135;
   8: 0, 2,  747,  29760,  359975,  1827210,   4369365,    4864860,    2027025;
   9: 0, 2, 1513,  97251, 1790195, 13758745,  51606555,   99324225,   93918825,   34459425;
  10: 0, 2, 3047, 309408, 8402108, 93553460, 514383870, 1516875360, 2435132700, 1998646650, 654729075;

%F Formula

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

%t Mathematica

P[1, x_] := 1 + x;
P[2, x_] := 2*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[(n - 2)*x*P[n - 2, x] + (x + x^2)*D[P[n - 2, x], x] - (1 - n*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + x; v[2] = 2*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = (n - 2)*x*v[i - 2] + (x + x^2)*deriv(v[i - 2], x) - (1 - n*x)*v[i - 1] + (x + 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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-0730b0b2fd7e59bf

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-0730b0b2fd7e59bf

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m),…
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check ma_wang_interval e-ade83b202f7a862b root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
- premise (sequence): Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m),…
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  
  no recorded property covers this requirement

Related sequences ?

Same fingerprint (1)

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

A134991 → Pₙ + Pₙ₊₁

References

[Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.