A059344 → basis A007318 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 8\,x + 4\,n + 8\,n x\right)\,P_{n-2}(x) + \left(1\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	5	8							13
3	13	24							37
4	73	240	192						505
5	281	1040	960						2281
6	1741	8760	14400	7680					32581
7	8485	47208	87360	53760					196813
8	57233	389984	981120	1075200	430080				2933617
9	328753	2443680	6797952	8386560	3870720				21827665
10	2389141	20603880	70197120	117734400	96768000	30965760			338658301
11	15539261	144651320	537609600	997032960	922521600	340623360			2957978101
12	120661465	1261466448	5439424320	12354693120	15540940800	10218700800	2724986880		47660873833
13	866545993	9696498864	45131211840	111822796800	155537141760	115130695680	35424829440		473609720377
14	7140942173	87841546520	459173787072	1319966968320	2248554147840	2262760980480	1239869030400	283398635520	7908706038325

Technical details

Node id: n-2a33317973f0fc37 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-24862e8f9dbb3493

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:      5,       8;
  3:     13,      24;
  4:     73,     240,     192;
  5:    281,    1040,     960;
  6:   1741,    8760,   14400,    7680;
  7:   8485,   47208,   87360,   53760;
  8:  57233,  389984,  981120, 1075200,  430080;
  9: 328753, 2443680, 6797952, 8386560, 3870720;

%F Formula

The row polynomials satisfy P(n,x) = (4*n-4+(8*n-8)*x)*P(n-2,x) + 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[(4*n - 4 + (8*n - 8)*x)*P[n - 2, x] + P[n - 1, x]];
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] = (4*n - 4 + (8*n - 8)*x)*v[i - 2] + v[i - 1]); 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 basis_map_A007318 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in t…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? 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 Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Parent-row provers of polytool (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