A107131 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-3 - x + x^{2} + 3\,n + n x - n x^{2}\right)\,P_{n-2}(x) + \left(-x + 2\,x^{2} + n x - 2\,n x^{2}\right)\,P_{n-2}'(x) + \left(x^{2} - n x^{2}\right)\,P_{n-2}''(x) + \left(1 + x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(x + 2\,n x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	2	4	1													7
3	4	16	9	1												30
4	9	63	63	16	1											152
5	21	251	400	175	25	1										873
6	51	1013	2431	1630	395	36	1									5557
7	127	4145	14462	13951	5075	777	49	1								38587
8	323	17181	85163	113638	58065	13174	1386	64	1							288995
9	835	72061	499422	897667	618051	194061	29988	2298	81	1						2314465
10	2188	305460	2926561	6953155	6271105	2610594	552657	61800	3600	100	1					19687221
11	5798	1307098	17170153	53176761	61563040	32997679	9159843	1392897	117810	5390	121	1				176896591
12	15511	5640297	100973665	403356668	590376611	398912118	140915698	27873846	3187932	210925	7777	144	1			1671471193
13	41835	24520341	595568376	3043666561	5566736175	4666856623	2052860524	512368714	75762258	6748170	358644	10881	169	1		16545499272
14	113634	107305630	3524374127	22895344199	51846109679	53266887674	28703245571	8841885910	1639069861	187846659	13391378	584038	14833	196	1	171026173390

Technical details

Node id: n-9302c5068cc7fd22 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-2bbe6df6dc024c8b

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:   1;
  1:   1,     1;
  2:   2,     4,      1;
  3:   4,    16,      9,      1;
  4:   9,    63,     63,     16,      1;
  5:  21,   251,    400,    175,     25,      1;
  6:  51,  1013,   2431,   1630,    395,     36,     1;
  7: 127,  4145,  14462,  13951,   5075,    777,    49,    1;
  8: 323, 17181,  85163, 113638,  58065,  13174,  1386,   64,  1;
  9: 835, 72061, 499422, 897667, 618051, 194061, 29988, 2298, 81, 1;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = (3*n-3+(n-1)*x-(n-1)*x^2)*P(n-2,x) + ((n-1)*x-(2*n-2)*x^2)*d/dx P(n-2,x) - (n-1)*x^2*d^2/dx^2 P(n-2,x) + (2*n+1+(2*n+1)*x)*P(n-1,x) + (2*n+1)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (n - 1)*x - (n - 1)*x^2)*P[n - 2, x] + ((n - 1)*x - (2*n - 2)*x^2)*D[P[n - 2, x], x] - (n - 1)*x^2*D[P[n - 2, x], {x, 2}] + (2*n + 1 + (2*n + 1)*x)*P[n - 1, x] + (2*n + 1)*x*D[P[n - 1, x], x])/(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 + x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (n - 1)*x - (n - 1)*x^2)*v[i - 2] + ((n - 1)*x - (2*n - 2)*x^2)*deriv(v[i - 2], x) - (n - 1)*x^2*deriv(deriv(v[i - 2], x), x) + (2*n + 1 + (2*n + 1)*x)*v[i - 1] + (2*n + 1)*x*deriv(v[i - 1], 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 (−∞, 0]
by Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from A Motzkin related triangle.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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
[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