A185263 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(1 - n - n^{2} - 4\,n^{2} x + n^{3} + 4\,n^{3} x\right)\,P_{n-2}(x) + \left(1 + 2\,n\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	2	4							6
3	6	20							26
4	24	124	128						276
5	120	812	1344						2276
6	720	6192	16144	11520					34576
7	5040	51984	176016	194816					427856
8	40320	490464	2120784	3732224	2064384				8448176
9	362880	5059296	26153424	59257088	49471488				140304176
10	3628800	57366720	350375904	1017052736	1359522816	619315200			3407262176
11	39916800	703609920	4917672864	17006406976	29033778176	19514327040			71215711776
12	479001600	9344298240	74111332608	303211846848	664887141376	719712681984	280284364800		2052030667456
13	6227020800	132971915520	1174847876352	5490919906752	14296001533952	19619675619328	11057859919872		51778503792576
14	87178291200	2026622868480	19823744014848	105201338187264	324400505684224	573123249168384	522906903576576	178541140377600	1726110682168576

Technical details

Node id: n-5e06d0aee5605183 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-057bf2a1d2daee1f

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:      2,       4;
  3:      6,      20;
  4:     24,     124,      128;
  5:    120,     812,     1344;
  6:    720,    6192,    16144,    11520;
  7:   5040,   51984,   176016,   194816;
  8:  40320,  490464,  2120784,  3732224,  2064384;
  9: 362880, 5059296, 26153424, 59257088, 49471488;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = (n^3-n^2-n+1+(4*n^3-4*n^2)*x)*P(n-2,x) + (2*n+1)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - n^2 - n + 1 + (4*n^3 - 4*n^2)*x)*P[n - 2, x] + (2*n + 1)*P[n - 1, 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; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 - n^2 - n + 1 + (4*n^3 - 4*n^2)*x)*v[i - 2] + (2*n + 1)*v[i - 1])/(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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t)…?
  - 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