A132166 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(\left(4 + n\right)\,P_{n}(x) = \left(42 + 2\,x + 6\,n\right)\,P_{n-2}(x) + \left(36\,x + x^{2}\right)\,P_{n-2}'(x) + \left(52 + 2\,x + 5\,n\right)\,P_{n-1}(x) + \left(36\,x + x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 4872 + 1155\,x + 64\,x^{2} + x^{3}\)

\(P_{1}(x) = 58968 + 24297\,x + 2394\,x^{2} + 85\,x^{3} + x^{4}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	P_n(1)
0	4872	1155	64	1								6092
1	58968	24297	2394	85	1							85745
2	653184	435456	67536	4074	106	1						1160357
3	6757344	6944112	1583253	143850	6195	127	1					15434882
4	66204864	101150208	32430888	4150629	262500	8757	148	1				204207995
5	620478144	1369540224	598274856	103312881	8980335	432747	11760	169	1			2701031117
6	5604318720	17451023616	10141103232	2292428736	262514826	17109603	663852	15204	190	1		35769177980
7	49063262976	211206393216	160202543904	46347017184	6787922661	573770925	29770146	965076	19089	211	1	474211665389

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-e8b8a3fcbab4370a · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-30085903e43caa64

Observed on the stored rows ?

Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
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:          4872,           1155,             64,              1;
  1:         58968,          24297,           2394,             85,             1;
  2:        653184,         435456,          67536,           4074,           106,            1;
  3:       6757344,        6944112,        1583253,         143850,          6195,          127,           1;
  4:      66204864,      101150208,       32430888,        4150629,        262500,         8757,         148,          1;
  5:     620478144,     1369540224,      598274856,      103312881,       8980335,       432747,       11760,        169,        1;
  6:    5604318720,    17451023616,    10141103232,     2292428736,     262514826,     17109603,      663852,      15204,      190,       1;
  7:   49063262976,   211206393216,   160202543904,    46347017184,    6787922661,    573770925,    29770146,     965076,    19089,     211,     1;
  8:  418184073216,  2445215205888,  2383504014528,   867056576256,  158881287240,  16915659111,  1125708948,   48388158,  1345680,   23415,   232,   1;
  9: 3482609872896, 27232873920000, 33668882356032, 15182785836288, 3422039202360, 449077969881, 37235476656, 2035934334, 74584314, 1814925, 28182, 253, 1;

%F Formula

The row polynomials satisfy (n+4)*P(n,x) = (6*n+42+2*x)*P(n-2,x) + (36*x+x^2)*d/dx P(n-2,x) + (5*n+52+2*x)*P(n-1,x) + (36*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 4872 + 1155*x + 64*x^2 + x^3, P(1,x) = 58968 + 24297*x + 2394*x^2 + 85*x^3 + x^4.

%t Mathematica

P[0, x_] := 4872 + 1155*x + 64*x^2 + x^3;
P[1, x_] := 58968 + 24297*x + 2394*x^2 + 85*x^3 + x^4;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 42 + 2*x)*P[n - 2, x] + (36*x + x^2)*D[P[n - 2, x], x] + (5*n + 52 + 2*x)*P[n - 1, x] + (36*x + x^2)*D[P[n - 1, x], x])/(n + 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4872 + 1155*x + 64*x^2 + x^3; v[2] = 58968 + 24297*x + 2394*x^2 + 85*x^3 + x^4; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 42 + 2*x)*v[i - 2] + (36*x + x^2)*deriv(v[i - 2], x) + (5*n + 52 + 2*x)*v[i - 1] + (36*x + x^2)*deriv(v[i - 1], x))/(n + 4)); 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-0a71e6005af9c857

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-0a71e6005af9c857

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): A convolution triangle of numbers obtained from A036224.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check ma_wang_interval e-fc9a4a17582197de root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
- premise (sequence): A convolution triangle of numbers obtained from A036224.
  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
[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.