A046716 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-16 + 8\,n\right)\,P_{n}(x) = \left(288 + 3864\,x + 17568\,x^{2} + 28800\,x^{3} + 7680\,x^{4} - 255\,n - 3568\,n x - 16800\,n x^{2} - 28416\,n x^{3} - 7936\,n x^{4} + 80\,n^{2} + 1096\,n^{2} x + 5088\,n^{2} x^{2} + 8576\,n^{2} x^{3} + 2560\,n^{2} x^{4} - 9\,n^{3} - 112\,n^{3} x - 480\,n^{3} x^{2} - 768\,n^{3} x^{3} - 256\,n^{3} x^{4}\right)\,P_{n-3}(x) + \left(24 + 408\,x + 2592\,x^{2} + 7296\,x^{3} + 7680\,x^{4} - 23\,n - 400\,n x - 2592\,n x^{2} - 7424\,n x^{3} - 7936\,n x^{4} + 8\,n^{2} + 136\,n^{2} x + 864\,n^{2} x^{2} + 2432\,n^{2} x^{3} + 2560\,n^{2} x^{4} - n^{3} - 16\,n^{3} x - 96\,n^{3} x^{2} - 256\,n^{3} x^{3} - 256\,n^{3} x^{4}\right)\,P_{n-3}'(x) + \left(-128 - 1032\,x - 2272\,x^{2} - 1280\,x^{3} + 117\,n + 1024\,n x + 2480\,n x^{2} + 1536\,n x^{3} - 30\,n^{2} - 276\,n^{2} x - 768\,n^{2} x^{2} - 576\,n^{2} x^{3} + n^{3} + 12\,n^{3} x + 48\,n^{3} x^{2} + 64\,n^{3} x^{3}\right)\,P_{n-2}(x) + \left(-8 - 136\,x - 736\,x^{2} - 1280\,x^{3} + 13\,n + 192\,n x + 944\,n x^{2} + 1536\,n x^{3} - 6\,n^{2} - 84\,n^{2} x - 384\,n^{2} x^{2} - 576\,n^{2} x^{3} + n^{3} + 12\,n^{3} x + 48\,n^{3} x^{2} + 64\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(32 + 184\,x - 160\,x^{2} - 43\,n - 208\,n x + 112\,n x^{2} + 15\,n^{2} + 56\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-8 - 72\,x - 160\,x^{2} + 5\,n + 48\,n x + 112\,n x^{2} - n^{2} - 8\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 6 + 4\,x\)

\(P_{3}(x) = 25 + 52\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	6	4							10
3	25	52	16						93
4	116	452	368	64					1000
5	641	4000	7088	2816	256				14801
6	4170	37572	114976	128960	29184	1024			315886
7	31209	376260	1703936	3483840	2898944	455680	4096		8953965
8	263800	4030292	24583728	75157760	116717824	78430208	9859072	16384	309059068

All 15 rows

Technical details

Node id: n-cf764ce34a69b783 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-7eb210ff9c608c87

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:        1;
   2:        6,         4;
   3:       25,        52,         16;
   4:      116,       452,        368,          64;
   5:      641,      4000,       7088,        2816,         256;
   6:     4170,     37572,     114976,      128960,       29184,         1024;
   7:    31209,    376260,    1703936,     3483840,     2898944,       455680,         4096;
   8:   263800,   4030292,   24583728,    75157760,   116717824,     78430208,      9859072,       16384;
   9:  2484545,  46259424,  357943776,  1475783168,  3434121984,   4333297664,   2463293440,   263979008,      65536;
  10: 25803886, 568896708, 5361331008, 28017332608, 87834886656, 165911006208, 177146937344, 87636934656, 8239054848, 262144;

%F Formula

The row polynomials satisfy (8*n-16)*P(n,x) = -(9*n^3-80*n^2+255*n-288+(112*n^3-1096*n^2+3568*n-3864)*x+(480*n^3-5088*n^2+16800*n-17568)*x^2+(768*n^3-8576*n^2+28416*n-28800)*x^3+(256*n^3-2560*n^2+7936*n-7680)*x^4)*P(n-3,x) - (n^3-8*n^2+23*n-24+(16*n^3-136*n^2+400*n-408)*x+(96*n^3-864*n^2+2592*n-2592)*x^2+(256*n^3-2432*n^2+7424*n-7296)*x^3+(256*n^3-2560*n^2+7936*n-7680)*x^4)*d/dx P(n-3,x) + (n^3-30*n^2+117*n-128+(12*n^3-276*n^2+1024*n-1032)*x+(48*n^3-768*n^2+2480*n-2272)*x^2+(64*n^3-576*n^2+1536*n-1280)*x^3)*P(n-2,x) + (n^3-6*n^2+13*n-8+(12*n^3-84*n^2+192*n-136)*x+(48*n^3-384*n^2+944*n-736)*x^2+(64*n^3-576*n^2+1536*n-1280)*x^3)*d/dx P(n-2,x) + (15*n^2-43*n+32+(56*n^2-208*n+184)*x-(16*n^2-112*n+160)*x^2)*P(n-1,x) - (n^2-5*n+8+(8*n^2-48*n+72)*x+(16*n^2-112*n+160)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 6 + 4*x, P(3,x) = 25 + 52*x + 16*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 6 + 4*x;
P[3, x_] := 25 + 52*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^3 - 80*n^2 + 255*n - 288 + (112*n^3 - 1096*n^2 + 3568*n - 3864)*x + (480*n^3 - 5088*n^2 + 16800*n - 17568)*x^2 + (768*n^3 - 8576*n^2 + 28416*n - 28800)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (16*n^3 - 136*n^2 + 400*n - 408)*x + (96*n^3 - 864*n^2 + 2592*n - 2592)*x^2 + (256*n^3 - 2432*n^2 + 7424*n - 7296)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*D[P[n - 3, x], x] + (n^3 - 30*n^2 + 117*n - 128 + (12*n^3 - 276*n^2 + 1024*n - 1032)*x + (48*n^3 - 768*n^2 + 2480*n - 2272)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (12*n^3 - 84*n^2 + 192*n - 136)*x + (48*n^3 - 384*n^2 + 944*n - 736)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*D[P[n - 2, x], x] + (15*n^2 - 43*n + 32 + (56*n^2 - 208*n + 184)*x - (16*n^2 - 112*n + 160)*x^2)*P[n - 1, x] - (n^2 - 5*n + 8 + (8*n^2 - 48*n + 72)*x + (16*n^2 - 112*n + 160)*x^2)*D[P[n - 1, x], x])/(8*n - 16)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 4*x; v[3] = 25 + 52*x + 16*x^2; for(n = 4, N, my(i = n); v[i] = (-(9*n^3 - 80*n^2 + 255*n - 288 + (112*n^3 - 1096*n^2 + 3568*n - 3864)*x + (480*n^3 - 5088*n^2 + 16800*n - 17568)*x^2 + (768*n^3 - 8576*n^2 + 28416*n - 28800)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (16*n^3 - 136*n^2 + 400*n - 408)*x + (96*n^3 - 864*n^2 + 2592*n - 2592)*x^2 + (256*n^3 - 2432*n^2 + 7424*n - 7296)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*deriv(v[i - 3], x) + (n^3 - 30*n^2 + 117*n - 128 + (12*n^3 - 276*n^2 + 1024*n - 1032)*x + (48*n^3 - 768*n^2 + 2480*n - 2272)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (12*n^3 - 84*n^2 + 192*n - 136)*x + (48*n^3 - 384*n^2 + 944*n - 736)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*deriv(v[i - 2], x) + (15*n^2 - 43*n + 32 + (56*n^2 - 208*n + 184)*x - (16*n^2 - 112*n + 160)*x^2)*v[i - 1] - (n^2 - 5*n + 8 + (8*n^2 - 48*n + 72)*x + (16*n^2 - 112*n + 160)*x^2)*deriv(v[i - 1], x))/(8*n - 16)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013611 of Coefficients of a special case of Poisson-Charlier polynomials.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Coefficients of a special case of Poisson-Charlier polynomials.?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] 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
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] 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
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Coefficients of a special case of Poisson-Charlier polynomials.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Coefficients of a special case of Poisson-Charlier polynomials.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Coefficients of a special case of Poisson-Charlier polynomials.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (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