A075498 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

\(P_{0}(x) = 4 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	4	1															5
1	12	10	1														23
2	36	72	19	1													128
3	108	468	243	31	1												851
4	324	2916	2655	615	46	1											6557
5	972	17820	26811	10035	1305	64	1										57008
6	2916	107892	259119	147231	29610	2457	85	1									549311
7	8748	650268	2439963	2025891	591381	73836	4242	109	1								5794439
8	26244	3910356	22609935	26750655	10896606	1920429	162918	6858	136	1							66284138
9	78732	23488380	207399771	343617795	190199745	45464328	5341707	327510	10530	166	1						815928665
10	236196	141009012	1890086319	4330813311	3196613970	1008557649	157640175	13201947	611820	15510	199	1					10738786109
11	708588	846290268	17151785883	53859846051	52280022861	21350651652	4319001324	474486903	29721087	1077120	22077	235	1				150313614050
12	2125764	5078450196	155212363215	663469938495	838060188966	436591752597	112049679456	15706686996	1276956252	62034687	1805661	30537	274	1			2227512013097
13	6377292	30472826940	1401989719131	8116851625155	13234372772985	8696711735712	2789635021173	489010167360	50184505800	3137996862	121621500	2904993	41223	316	1		34812497316443
14	19131876	182843338932	12648380299119	98804209220991	206632443219930	169775184015801	67279047180345	14525879037813	1843991823960	144324411660	7151506362	226201248	4512690	54495	361	1	571843703955584

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Technical details

Node id: n-b3e09542618124d6 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-9b67f6049f522494

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:     4,        1;
  1:    12,       10,         1;
  2:    36,       72,        19,         1;
  3:   108,      468,       243,        31,         1;
  4:   324,     2916,      2655,       615,        46,        1;
  5:   972,    17820,     26811,     10035,      1305,       64,       1;
  6:  2916,   107892,    259119,    147231,     29610,     2457,      85,      1;
  7:  8748,   650268,   2439963,   2025891,    591381,    73836,    4242,    109,     1;
  8: 26244,  3910356,  22609935,  26750655,  10896606,  1920429,  162918,   6858,   136,   1;
  9: 78732, 23488380, 207399771, 343617795, 190199745, 45464328, 5341707, 327510, 10530, 166, 1;

%F Formula

The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + x.

%t Mathematica

P[0, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + 3*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] = 4 + x; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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 - 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-53ebd0bab53d0b02

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-53ebd0bab53d0b02

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Stirling2 triangle with scaled diagonals (powers of 3).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-5cf851f11a600529 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Stirling2 triangle with scaled diagonals (powers of 3).
  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.