A225476 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	1								3
1	2	5	2							9
2	2	17	20	6						45
3	2	53	134	102	24					315
4	2	161	776	1116	624	120				2799
5	2	485	4202	10140	10080	4440	720			30069
6	2	1457	21980	83586	131280	99240	36000	5040		378585
7	2	4373	112814	651042	1515864	1748040	1063440	327600	40320	5463495

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-d519174be8286cb2 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-484d0c9ef1fc7fef

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,       2;
  2: 2,    17,      20,        6;
  3: 2,    53,     134,      102,        24;
  4: 2,   161,     776,     1116,       624,       120;
  5: 2,   485,    4202,    10140,     10080,      4440,       720;
  6: 2,  1457,   21980,    83586,    131280,     99240,     36000,      5040;
  7: 2,  4373,  112814,   651042,   1515864,   1748040,   1063440,    327600,     40320;
  8: 2, 13121,  572816,  4895736,  16246944,  26807760,  24312960,  12358080,   3306240,   362880;
  9: 2, 39365, 2890322, 35988600, 165805440, 376120080, 476915040, 355561920, 155070720, 36650880, 3628800;

%F Formula

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

%t Mathematica

P[0, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (2*x + 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; for(n = 1, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (2*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 - 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-b0c8196dbc01af81

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-b0c8196dbc01af81

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check ma_wang_interval e-0759b614e8dc4111 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
- premise (sequence): Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k…
  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
[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.