A035607 → basis A007318 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(8 + 6\,x - 14\,x^{2} - 14\,x^{3} - 2\,x^{4} - 8\,n - 24\,n x - 16\,n x^{2} + 4\,n x^{3} + n x^{4} + 2\,n^{2} x + 8\,n^{2} x^{2} + n^{2} x^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-8 - 6\,x + 14\,x^{2} + 14\,x^{3} + 2\,x^{4} + 16\,n + 22\,n x - 12\,n x^{2} - 20\,n x^{3} - 3\,n x^{4} - 24\,n^{2} x - 24\,n^{2} x^{2} - 3\,n^{2} x^{3} - 8\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(32 + 152\,x + 276\,x^{2} + 239\,x^{3} + 99\,x^{4} + 17\,x^{5} + x^{6} + 48\,n + 160\,n x + 190\,n x^{2} + 94\,n x^{3} + 17\,n x^{4} + n x^{5} + 16\,n^{2} + 32\,n^{2} x + 18\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	1														1
2	5	2													7
3	15	12	2												29
4	45	56	20	2											123
5	129	220	126	28	2										505
6	361	786	628	226	36	2									2039
7	991	2632	2716	1376	356	44	2								8117
8	2681	8416	10680	7036	2572	516	52	2							31955
9	7169	25992	39220	31944	15240	4324	706	60	2						124657
10	18989	78130	136792	133036	78724	29200	6740	926	68	2					482607
11	49903	229812	458286	518912	367692	169476	51166	9928	1176	76	2				1856429
12	130277	664056	1486572	1922542	1589892	879084	330108	83766	13996	1456	84	2			7101835
13	338177	1890564	4696002	6833572	6468114	4180908	1885422	595276	130006	19052	1766	92	2		27038953
14	873553	5315170	14509740	23474270	25045760	18557802	9794460	3715850	1009680	193270	25204	2106	100	2	102516967

Technical details

Node id: n-96da891f206ebda7 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-b074c357d42fbaf1

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    1;
  1:    1;
  2:    5,     2;
  3:   15,    12,     2;
  4:   45,    56,    20,     2;
  5:  129,   220,   126,    28,     2;
  6:  361,   786,   628,   226,    36,    2;
  7:  991,  2632,  2716,  1376,   356,   44,   2;
  8: 2681,  8416, 10680,  7036,  2572,  516,  52,  2;
  9: 7169, 25992, 39220, 31944, 15240, 4324, 706, 60, 2;

%F Formula

The row polynomials satisfy -(8*n-8-(2*n^3+2*n^2-24*n+6)*x-(8*n^2-16*n-14)*x^2-(n^2+4*n-14)*x^3-(n-2)*x^4)*P(n,x) = -(8*n^3-16*n+8+(6*n^3+24*n^2-22*n+6)*x+(24*n^2+12*n-14)*x^2+(3*n^2+20*n-14)*x^3+(3*n-2)*x^4)*P(n-1,x) + (16*n^2+48*n+32+(32*n^2+160*n+152)*x+(18*n^2+190*n+276)*x^2+(2*n^2+94*n+239)*x^3+(17*n+99)*x^4+(n+17)*x^5+x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 16*n + 8 + (6*n^3 + 24*n^2 - 22*n + 6)*x + (24*n^2 + 12*n - 14)*x^2 + (3*n^2 + 20*n - 14)*x^3 + (3*n - 2)*x^4)*P[n - 1, x] + (16*n^2 + 48*n + 32 + (32*n^2 + 160*n + 152)*x + (18*n^2 + 190*n + 276)*x^2 + (2*n^2 + 94*n + 239)*x^3 + (17*n + 99)*x^4 + (n + 17)*x^5 + x^6)*D[P[n - 1, x], x])/(-8*n + 8 + (2*n^3 + 2*n^2 - 24*n + 6)*x + (8*n^2 - 16*n - 14)*x^2 + (n^2 + 4*n - 14)*x^3 + (n - 2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(8*n^3 - 16*n + 8 + (6*n^3 + 24*n^2 - 22*n + 6)*x + (24*n^2 + 12*n - 14)*x^2 + (3*n^2 + 20*n - 14)*x^3 + (3*n - 2)*x^4)*v[i - 1] + (16*n^2 + 48*n + 32 + (32*n^2 + 160*n + 152)*x + (18*n^2 + 190*n + 276)*x^2 + (2*n^2 + 94*n + 239)*x^3 + (17*n + 99)*x^4 + (n + 17)*x^5 + x^6)*deriv(v[i - 1], x))/(-8*n + 8 + (2*n^3 + 2*n^2 - 24*n + 6)*x + (8*n^2 - 16*n - 14)*x^2 + (n^2 + 4*n - 14)*x^3 + (n - 2)*x^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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A007318 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercu…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

References

[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.