A035342 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(\left(6 - x^{2} + 22\,n + 24\,n^{2} + 8\,n^{3}\right)\,P_{n}(x) = \left(12 + 8\,x - 3\,x^{2} - x^{3} + 62\,n + 26\,n x - 2\,n x^{2} + 108\,n^{2} + 24\,n^{2} x + 72\,n^{3} + 8\,n^{3} x + 16\,n^{4}\right)\,P_{n-1}(x) + \left(8\,x - 4\,x^{2} - x^{3} + 26\,n x + 24\,n^{2} x + 8\,n^{3} x\right)\,P_{n-1}'(x) + \left(-2\,x^{2}\right)\,P_{n-1}''(x)\)

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

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	18	10	1														29
2	120	96	19	1													236
3	1050	1062	303	31	1												2447
4	11340	13620	4965	735	46	1											30707
5	145530	200070	87525	16665	1515	64	1										451370
6	2162160	3321360	1678635	386085	45360	2793	85	1									7596479
7	36486450	61571790	35049735	9368415	1335705	106680	4746	109	1								143923631
8	689188500	1261654380	794424645	240414615	40001850	3890565	225162	7578	136	1							3029807432
9	14404039650	28328709150	19466513325	6546378825	1237877235	140948640	9960615	437130	11520	166	1						70134876257
10	329983453800	691886626200	513469600875	189209530125	39891644100	5176865925	429365475	23059575	793980	16830	199	1					1770070957085
11	8222087723850	18264784749750	14518154764575	5798706519375	1343668874625	194923297800	18467715420	1166412555	49238145	1365870	23793	235	1				48362010685994
12	221363900257500	518306538449700	438301744805925	188113561210575	47375056678950	7571134996425	803518655940	58080482460	2888182440	98376135	2245815	32721	274	1			1421896524374861
13	6403741400306250	15735360006516150	14077613846020725	6446704079640825	1748880478820475	304413879369600	35658191774205	2892128459220	164846321640	6623777160	185915730	3554187	43953	316	1		44755435670520437
14	198089067316140000	508933848579156000	479448060217000875	232900557055291125	67565380677795000	12693017633321025	1622196033883425	145445199824925	9315455549400	429603354180	14241877650	335126610	5443620	57855	361	1	1501407332353822052

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Technical details

Node id: n-75126c0548fd070f · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-2a27715222e8de49

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:          18,          10,           1;
  2:         120,          96,          19,          1;
  3:        1050,        1062,         303,         31,          1;
  4:       11340,       13620,        4965,        735,         46,         1;
  5:      145530,      200070,       87525,      16665,       1515,        64,       1;
  6:     2162160,     3321360,     1678635,     386085,      45360,      2793,      85,      1;
  7:    36486450,    61571790,    35049735,    9368415,    1335705,    106680,    4746,    109,     1;
  8:   689188500,  1261654380,   794424645,  240414615,   40001850,   3890565,  225162,   7578,   136,   1;
  9: 14404039650, 28328709150, 19466513325, 6546378825, 1237877235, 140948640, 9960615, 437130, 11520, 166, 1;

%F Formula

The row polynomials satisfy (8*n^3+24*n^2+22*n+6-x^2)*P(n,x) = (16*n^4+72*n^3+108*n^2+62*n+12+(8*n^3+24*n^2+26*n+8)*x-(2*n+3)*x^2-x^3)*P(n-1,x) + ((8*n^3+24*n^2+26*n+8)*x-4*x^2-x^3)*d/dx P(n-1,x) - 2*x^2*d^2/dx^2 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[Together[((16*n^4 + 72*n^3 + 108*n^2 + 62*n + 12 + (8*n^3 + 24*n^2 + 26*n + 8)*x - (2*n + 3)*x^2 - x^3)*P[n - 1, x] + ((8*n^3 + 24*n^2 + 26*n + 8)*x - 4*x^2 - x^3)*D[P[n - 1, x], x] - 2*x^2*D[P[n - 1, x], {x, 2}])/(8*n^3 + 24*n^2 + 22*n + 6 - x^2)]];
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] = ((16*n^4 + 72*n^3 + 108*n^2 + 62*n + 12 + (8*n^3 + 24*n^2 + 26*n + 8)*x - (2*n + 3)*x^2 - x^3)*v[i - 1] + ((8*n^3 + 24*n^2 + 26*n + 8)*x - 4*x^2 - x^3)*deriv(v[i - 1], x) - 2*x^2*deriv(deriv(v[i - 1], x), x))/(8*n^3 + 24*n^2 + 22*n + 6 - x^2)); 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-91082a28c5d8ee02

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-91082a28c5d8ee02

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): The convolution matrix of the double factorial of odd numbers (A001147).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-cf2bc165f5c18b21 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): The convolution matrix of the double factorial of odd numbers (A001147).
  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.