A103327 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(3 - 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(4 + 12\,x - 12\,n - 39\,n x + 8\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} - 4\,n x - 12\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	3														3
2	36	18													54
3	240	432	81												753
4	1344	4320	3240	324											9228
5	6912	32256	45360	19440	1215										105183
6	33792	207360	435456	362880	102060	4374									1145922
7	159744	1216512	3421440	4354560	2449440	489888	15309								12106893
8	737280	6709248	23721984	41057280	35925120	14696640	2204496	52488							125104536
9	3342336	35389440	150958080	332107776	400308480	258660864	80831520	9447840	177147						1271223483
10	14942208	180486144	902430720	2415329280	3736212480	3362591232	1681295616	415704960	38972340	590490					12748555470
11	66060288	896532480	5143855104	16243752960	30795448320	35867639808	25219434240	10087773696	2026561680	155889360	1948617				126504896553
12	289406976	4359979008	28240773120	102877102080	231473479680	332590841856	304874938368	172933263360	56743727040	9457287840	607968504	6377292			1244455145124
13	1258291200	20837302272	150419275776	621297008640	1620314357760	2777681756160	3159612997632	2351892381696	1102449553920	302633210880	42557795280	2321334288	20726199		12153295991703
14	5435817984	98146713600	781398835200	3610062618624	10717373399040	21388149522432	29165658439680	27082397122560	16757233219584	6614697323520	1543429375488	185706743040	8705003580	66961566	117958461095898

Technical details

Node id: n-3acc34bd3f89fd5a · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-60f97f8d0ae4f0f4

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
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:       0;
  1:       3;
  2:      36,       18;
  3:     240,      432,        81;
  4:    1344,     4320,      3240,       324;
  5:    6912,    32256,     45360,     19440,      1215;
  6:   33792,   207360,    435456,    362880,    102060,      4374;
  7:  159744,  1216512,   3421440,   4354560,   2449440,    489888,    15309;
  8:  737280,  6709248,  23721984,  41057280,  35925120,  14696640,  2204496,   52488;
  9: 3342336, 35389440, 150958080, 332107776, 400308480, 258660864, 80831520, 9447840, 177147;

%F Formula

The row polynomials satisfy (2*n^2-5*n+3)*P(n,x) = (8*n^2-12*n+4+(18*n^2-39*n+12)*x)*P(n-1,x) - ((4*n-2)*x+(12*n-6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 12*n + 4 + (18*n^2 - 39*n + 12)*x)*P[n - 1, x] - ((4*n - 2)*x + (12*n - 6)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 12*n + 4 + (18*n^2 - 39*n + 12)*x)*v[i - 1] - ((4*n - 2)*x + (12*n - 6)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 3)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023