A225472 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

\(P_{1}(x) = 3 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	3	3								6
2	6	24	18							48
3	12	138	288	162						600
4	24	726	3132	4374	1944					10200
5	48	3702	29412	76302	79704	29160				218328
6	96	18654	257508	1104030	2031480	1691280	524880			5627928
7	192	93558	2171988	14461902	41689080	59223960	40940640	11022480		169603800
8	384	468366	17937252	178628814	757189944	1632143520	1884844080	1113270480	264539520	5849022360

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-464957f72336fbd0 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-24e672df326049ba

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:    3,        3;
   2:    6,       24,         18;
   3:   12,      138,        288,         162;
   4:   24,      726,       3132,        4374,         1944;
   5:   48,     3702,      29412,       76302,        79704,        29160;
   6:   96,    18654,     257508,     1104030,      2031480,      1691280,        524880;
   7:  192,    93558,    2171988,    14461902,     41689080,     59223960,      40940640,      11022480;
   8:  384,   468366,   17937252,   178628814,    757189944,   1632143520,    1884844080,    1113270480,     264539520;
   9:  768,  2342982,  146308212,  2126352222,  12744204984,  39104289000,   67075464960,   65186946720,   33596519040,    7142567040;
  10: 1536, 11717214, 1184523588, 24706648350, 203935096440, 855935987760, 2045386501200, 2907884538720, 2437996216320, 1114240458240, 214277011200;

%F Formula

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

%t Mathematica

P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 3*x)*P[n - 1, x] + (3*x + 3*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] = 3 + 3*x; for(n = 2, N, my(i = n); v[i] = (2 + 3*x)*v[i - 1] + (3*x + 3*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-2dc56bf9965267b7

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-2dc56bf9965267b7

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, k!*S_3(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 >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check ma_wang_interval e-2d30e6aff83a7d44 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
- premise (sequence): Triangle read by rows, k!*S_3(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 >= 2
  
  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.