A111577 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

\(P_{1}(x) = 2 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	2	1															3
2	2	6	1														9
3	2	26	13	1													42
4	2	106	117	23	1												249
5	2	426	925	347	36	1											1737
6	2	1706	6901	4395	815	52	1										13872
7	2	6826	50013	50851	14990	1647	71	1									124401
8	2	27306	356917	558523	245721	41342	2996	93	1								1232901
9	2	109226	2525725	5942147	3752896	907193	98266	5042	118	1							13340616
10	2	436906	17789301	61947195	54729795	18267984	2774247	209190	7992	146	1						156162759
11	2	1747626	124962013	637261251	773434530	347017539	70978677	7376427	408990	12080	177	1					1963199313
12	2	6990506	876481717	6497574523	10691910141	6325715154	1695612402	233260071	17601177	747230	17567	211	1				26345910702
13	2	27962026	6142362525	65852226947	145492406356	111903352605	38542350792	6827333964	673289496	38523617	1291807	24741	248	1			375501125127
14	2	111848106	43024499701	664664631995	1957253509575	1935946048036	844208017653	188743698000	23659571364	1751950772	78569634	2133001	33917	288	1		5659444512045
15	2	447392426	301283346013	6689670819651	26108960256470	32932390278151	17975898383443	4996569373653	780232982100	72714192980	4187609426	151091668	3387930	45437	331	1	89862509159682

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Technical details

Node id: n-288ee4e6a60f9c49 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-08790148ebd3c793

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: 2,      1;
   2: 2,      6,        1;
   3: 2,     26,       13,        1;
   4: 2,    106,      117,       23,        1;
   5: 2,    426,      925,      347,       36,        1;
   6: 2,   1706,     6901,     4395,      815,       52,       1;
   7: 2,   6826,    50013,    50851,    14990,     1647,      71,      1;
   8: 2,  27306,   356917,   558523,   245721,    41342,    2996,     93,    1;
   9: 2, 109226,  2525725,  5942147,  3752896,   907193,   98266,   5042,  118,   1;
  10: 2, 436906, 17789301, 61947195, 54729795, 18267984, 2774247, 209190, 7992, 146, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 3*x*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] = 2 + x; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + 3*x*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-45f39b72ccd15e60

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-45f39b72ccd15e60

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-94f85f79154d7ed6 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.
  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
[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.