A062145 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(\left(-24\,n + 24\,n x - 6\,n x^{2} + 4\,n^{2} + 2\,n^{2} x + n^{2} x^{2} + 16\,n^{3} - 2\,n^{3} x + 4\,n^{3} x^{2} + 4\,n^{4} + n^{4} x^{2}\right)\,P_{n}(x) = \left(-24\,x + 36\,x^{2} - 12\,x^{3} - 24\,n + 8\,n x + 36\,n x^{2} - 8\,n x^{3} + 4\,n^{2} + 18\,n^{2} x + 21\,n^{2} x^{2} + 7\,n^{2} x^{3} + 16\,n^{3} + 34\,n^{3} x + 2\,n^{3} x^{2} + 10\,n^{3} x^{3} + 4\,n^{4} + 12\,n^{4} x + n^{4} x^{2} + 3\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(-12\,x + 36\,x^{2} - 30\,x^{3} + 6\,x^{4} + 4\,n x + 18\,n x^{2} - 23\,n x^{3} + n x^{4} + 24\,n^{2} x - 22\,n^{2} x^{2} + 3\,n^{2} x^{3} - 5\,n^{2} x^{4} + 8\,n^{3} x - 8\,n^{3} x^{2} + 2\,n^{3} x^{3} - 2\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	2	4															6
2	2	14	10														26
3	2	28	55	20													105
4	2	46	171	160	35												414
5	2	68	406	700	385	56											1617
6	2	94	820	2240	2240	812	84										6292
7	2	124	1485	5880	9240	6048	1554	120									24453
8	2	158	2485	13440	30450	31164	14406	2760	165								95030
9	2	196	3916	27720	85470	125664	90552	31152	4620	220							369512
10	2	238	5886	52800	212520	424116	437976	234432	62370	7370	286						1437996
11	2	284	8515	94380	480480	1249248	1747746	1338480	553410	117260	11297	364					5601466
12	2	334	11935	160160	1006005	3303300	6012006	6229080	3680820	1211210	209209	16744	455				21841260
13	2	388	16290	260260	1976975	7999992	18366348	24727560	19749015	9280700	2488486	357084	24115	560			85247775
14	2	446	21736	407680	3683680	18009992	50906856	86376576	89566620	56848220	21757736	4845568	586768	33880	680		333046440
15	2	508	28441	618800	6559280	38118080	130077948	271532976	354585660	292011720	150872722	47944624	9009728	932960	46580	816	1302340845

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Technical details

Node id: n-b7ab2dc5f7577856 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-27b0ef728bca3ff5

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,   4;
   2: 2,  14,   10;
   3: 2,  28,   55,    20;
   4: 2,  46,  171,   160,     35;
   5: 2,  68,  406,   700,    385,     56;
   6: 2,  94,  820,  2240,   2240,    812,     84;
   7: 2, 124, 1485,  5880,   9240,   6048,   1554,    120;
   8: 2, 158, 2485, 13440,  30450,  31164,  14406,   2760,   165;
   9: 2, 196, 3916, 27720,  85470, 125664,  90552,  31152,  4620,  220;
  10: 2, 238, 5886, 52800, 212520, 424116, 437976, 234432, 62370, 7370, 286;

%F Formula

The row polynomials satisfy (4*n^4+16*n^3+4*n^2-24*n-(2*n^3-2*n^2-24*n)*x+(n^4+4*n^3+n^2-6*n)*x^2)*P(n,x) = (4*n^4+16*n^3+4*n^2-24*n+(12*n^4+34*n^3+18*n^2+8*n-24)*x+(n^4+2*n^3+21*n^2+36*n+36)*x^2+(3*n^4+10*n^3+7*n^2-8*n-12)*x^3)*P(n-1,x) + ((8*n^3+24*n^2+4*n-12)*x-(8*n^3+22*n^2-18*n-36)*x^2+(2*n^3+3*n^2-23*n-30)*x^3-(2*n^3+5*n^2-n-6)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2 + 4*x.

%t Mathematica

P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^4 + 16*n^3 + 4*n^2 - 24*n + (12*n^4 + 34*n^3 + 18*n^2 + 8*n - 24)*x + (n^4 + 2*n^3 + 21*n^2 + 36*n + 36)*x^2 + (3*n^4 + 10*n^3 + 7*n^2 - 8*n - 12)*x^3)*P[n - 1, x] + ((8*n^3 + 24*n^2 + 4*n - 12)*x - (8*n^3 + 22*n^2 - 18*n - 36)*x^2 + (2*n^3 + 3*n^2 - 23*n - 30)*x^3 - (2*n^3 + 5*n^2 - n - 6)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 16*n^3 + 4*n^2 - 24*n - (2*n^3 - 2*n^2 - 24*n)*x + (n^4 + 4*n^3 + n^2 - 6*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2 + 4*x; for(n = 2, N, my(i = n); v[i] = ((4*n^4 + 16*n^3 + 4*n^2 - 24*n + (12*n^4 + 34*n^3 + 18*n^2 + 8*n - 24)*x + (n^4 + 2*n^3 + 21*n^2 + 36*n + 36)*x^2 + (3*n^4 + 10*n^3 + 7*n^2 - 8*n - 12)*x^3)*v[i - 1] + ((8*n^3 + 24*n^2 + 4*n - 12)*x - (8*n^3 + 22*n^2 - 18*n - 36)*x^2 + (2*n^3 + 3*n^2 - 23*n - 30)*x^3 - (2*n^3 + 5*n^2 - n - 6)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 16*n^3 + 4*n^2 - 24*n - (2*n^3 - 2*n^2 - 24*n)*x + (n^4 + 4*n^3 + n^2 - 6*n)*x^2)); 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-a6e517cc00955d8a

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-a6e517cc00955d8a

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-62efc507a1cd208a root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…
  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.