A035607 → basis A007318 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 - 5\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(47 + 31\,x + 4\,x^{2} - 17\,n - 10\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(34 + 37\,x + 12\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)

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

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

\(P_{2}(x) = 5 + 2\,x\)

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	17	12	2												31
4	61	62	20	2											145
5	217	288	138	28	2										673
6	773	1266	804	246	36	2									3127
7	2753	5364	4242	1736	386	44	2								14527
8	9805	22150	20964	10746	3212	558	52	2							67489
9	34921	89736	98890	60916	22890	5360	762	60	2						313537
10	124373	358234	450484	324094	146756	43298	8308	998	68	2					1456615
11	442961	1413468	1997202	1643488	871058	310260	75106	12184	1266	76	2				6767071
12	1577629	5524206	8664276	8026338	4874268	2035190	595492	121962	17116	1566	84	2			31438129
13	5618809	21420144	36924906	38027468	26034746	12471416	4282138	1060852	188026	23232	1898	92	2		146053729
14	20011685	82505282	155047620	175724262	133906580	72393642	28544004	8304110	1781124	277970	30660	2262	100	2	678529303

Technical details

Node id: n-d87adca1089fcb13 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-48a53241c969ef5c

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:    17,    12,     2;
  4:    61,    62,    20,     2;
  5:   217,   288,   138,    28,     2;
  6:   773,  1266,   804,   246,    36,    2;
  7:  2753,  5364,  4242,  1736,   386,   44,   2;
  8:  9805, 22150, 20964, 10746,  3212,  558,  52,  2;
  9: 34921, 89736, 98890, 60916, 22890, 5360, 762, 60, 2;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n^2-5*n+2)*P(n-1,x) - (17*n-47+(10*n-31)*x+(n-4)*x^2)*d/dx P(n-1,x) + (34+37*x+12*x^2+x^3)*d^2/dx^2 P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 5 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 2)*P[n - 1, x] - (17*n - 47 + (10*n - 31)*x + (n - 4)*x^2)*D[P[n - 1, x], x] + (34 + 37*x + 12*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 5 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 2)*v[i - 1] - (17*n - 47 + (10*n - 31)*x + (n - 4)*x^2)*deriv(v[i - 1], x) + (34 + 37*x + 12*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n - 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary), Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) 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)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from shift_x_plus_one of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercub…?
  - Affine, reversal and Möbius substitutions (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)
  - Affine, reversal and Möbius substitutions (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)

Related ?

Candidate equalities ?

A035607 → basis A007318 → basis A007318 15 rows; no all-index equality certificate was found

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