A049374 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(4 + x + n\right)\,P_{n-2}(x) + \left(5\,x\right)\,P_{n-2}'(x) + \left(5 + x + n\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 7 + x\)

\(P_{1}(x) = 48 + 19\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	7	1															8
1	48	19	1														68
2	378	294	37	1													710
3	3360	4476	996	61	1												8894
4	33264	70224	24360	2520	91	1											130460
5	362880	1152648	580824	89820	5340	127	1										2191640
6	4324320	19891872	13907376	3003504	260610	10038	169	1									41397890
7	55883520	361291392	339204096	97918128	11598384	641928	17304	217	1								866554970
8	778377600	6901614720	8488839744	3175913664	492007824	36623664	1403136	27936	271	1							19874808560
9	11623772160	138468185856	218796754176	103671247680	20384844480	1956322368	99747648	2799720	42840	331	1						495003717260
10	185253868800	2912896546560	5819020808832	3429524520576	837044984160	100558069920	6543328176	242505648	5198490	63030	397	1					13291089894590
11	3138418483200	64137068835840	159815824616448	115463215131264	34380284754816	5058563590080	407996132544	19150849872	538777008	9106020	89628	469	1				382421070367190
12	56317176115200	1475510014218240	4533371771056128	3966772989763584	1421096895009792	251799881389056	24635932985664	1422751442112	50394760416	1112393568	15200328	123864	547	1			11730978934458500
13	1067062284288000	35407201436006400	132789083937269760	139291519142071296	59355988751195136	12495331750277376	1458559046520576	101445599743488	4395508685568	121578905448	2161126968	24365796	167076	631	1		381970711220623520
14	21287892571545600	884832944251576320	4014700555220619264	5004137615200948224	2512143535438817280	621399077558138880	85399049694486528	7036663192086528	365126259738240	12297034189440	272842682112	3988438272	37731330	220710	721	1	13151315033291219450

Computed in your browser ?

Technical details

Node id: n-6ab27086b192524d · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-f99f37741a71f2e9

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:           7,            1;
  1:          48,           19,            1;
  2:         378,          294,           37,            1;
  3:        3360,         4476,          996,           61,           1;
  4:       33264,        70224,        24360,         2520,          91,          1;
  5:      362880,      1152648,       580824,        89820,        5340,        127,        1;
  6:     4324320,     19891872,     13907376,      3003504,      260610,      10038,      169,       1;
  7:    55883520,    361291392,    339204096,     97918128,    11598384,     641928,    17304,     217,     1;
  8:   778377600,   6901614720,   8488839744,   3175913664,   492007824,   36623664,  1403136,   27936,   271,   1;
  9: 11623772160, 138468185856, 218796754176, 103671247680, 20384844480, 1956322368, 99747648, 2799720, 42840, 331, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n+4+x)*P(n-2,x) + 5*x*d/dx P(n-2,x) + (n+5+x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 7 + x, P(1,x) = 48 + 19*x + x^2.

%t Mathematica

P[0, x_] := 7 + x;
P[1, x_] := 48 + 19*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n + 4 + x)*P[n - 2, x] + 5*x*D[P[n - 2, x], x] + (n + 5 + x)*P[n - 1, x] + 5*x*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] = 7 + x; v[2] = 48 + 19*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (n + 4 + x)*v[i - 2] + 5*x*deriv(v[i - 2], x) + (n + 5 + x)*v[i - 1] + 5*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 - 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-0a2b33f6b07eac20

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-0a2b33f6b07eac20

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): A triangle of numbers related to triangle A030527.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-eb06c12f54622221 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): A triangle of numbers related to triangle A030527.
  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.