A046716 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-8 + 4\,n\right)\,P_{n}(x) = \left(156 + 1068\,x + 2520\,x^{2} + 2256\,x^{3} + 480\,x^{4} - 139\,n - 992\,n x - 2424\,n x^{2} - 2240\,n x^{3} - 496\,n x^{4} + 44\,n^{2} + 308\,n^{2} x + 744\,n^{2} x^{2} + 688\,n^{2} x^{3} + 160\,n^{2} x^{4} - 5\,n^{3} - 32\,n^{3} x - 72\,n^{3} x^{2} - 64\,n^{3} x^{3} - 16\,n^{3} x^{4}\right)\,P_{n-3}(x) + \left(24 + 204\,x + 648\,x^{2} + 912\,x^{3} + 480\,x^{4} - 23\,n - 200\,n x - 648\,n x^{2} - 928\,n x^{3} - 496\,n x^{4} + 8\,n^{2} + 68\,n^{2} x + 216\,n^{2} x^{2} + 304\,n^{2} x^{3} + 160\,n^{2} x^{4} - n^{3} - 8\,n^{3} x - 24\,n^{3} x^{2} - 32\,n^{3} x^{3} - 16\,n^{3} x^{4}\right)\,P_{n-3}'(x) + \left(-68 - 292\,x - 376\,x^{2} - 160\,x^{3} + 65\,n + 304\,n x + 428\,n x^{2} + 192\,n x^{3} - 18\,n^{2} - 90\,n^{2} x - 144\,n^{2} x^{2} - 72\,n^{2} x^{3} + n^{3} + 6\,n^{3} x + 12\,n^{3} x^{2} + 8\,n^{3} x^{3}\right)\,P_{n-2}(x) + \left(-8 - 68\,x - 184\,x^{2} - 160\,x^{3} + 13\,n + 96\,n x + 236\,n x^{2} + 192\,n x^{3} - 6\,n^{2} - 42\,n^{2} x - 96\,n^{2} x^{2} - 72\,n^{2} x^{3} + n^{3} + 6\,n^{3} x + 12\,n^{3} x^{2} + 8\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(12 + 28\,x - 40\,x^{2} - 19\,n - 40\,n x + 28\,n x^{2} + 7\,n^{2} + 12\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-8 - 36\,x - 40\,x^{2} + 5\,n + 24\,n x + 28\,n x^{2} - n^{2} - 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{3}(x) = 15 + 18\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	4	2							6
3	15	18	4						37
4	66	138	68	8					280
5	353	1144	1100	288	16				2901
6	2248	10354	16480	10104	1664	32			40882
7	16583	101378	234680	249624	113824	13856	64		730009
8	138750	1070098	3309228	5170144	4175504	1537408	153152	128	15554412

All 15 rows

Technical details

Node id: n-2038bae4c1bb9757 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-cb11c1d5b8bd540c

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:        4,         2;
   3:       15,        18,         4;
   4:       66,       138,        68,          8;
   5:      353,      1144,      1100,        288,         16;
   6:     2248,     10354,     16480,      10104,       1664,         32;
   7:    16583,    101378,    234680,     249624,     113824,      13856,         64;
   8:   138750,   1070098,   3309228,    5170144,    4175504,    1537408,     153152,       128;
   9:  1297073,  12158592,  47485752,   99236480,  117970864,   77373952,   24081280,   2060288,      256;
  10: 13395148, 148428546, 704171328, 1857110192, 2950472256, 2846280288, 1578830336, 427266816, 32179200, 512;

%F Formula

The row polynomials satisfy (4*n-8)*P(n,x) = -(5*n^3-44*n^2+139*n-156+(32*n^3-308*n^2+992*n-1068)*x+(72*n^3-744*n^2+2424*n-2520)*x^2+(64*n^3-688*n^2+2240*n-2256)*x^3+(16*n^3-160*n^2+496*n-480)*x^4)*P(n-3,x) - (n^3-8*n^2+23*n-24+(8*n^3-68*n^2+200*n-204)*x+(24*n^3-216*n^2+648*n-648)*x^2+(32*n^3-304*n^2+928*n-912)*x^3+(16*n^3-160*n^2+496*n-480)*x^4)*d/dx P(n-3,x) + (n^3-18*n^2+65*n-68+(6*n^3-90*n^2+304*n-292)*x+(12*n^3-144*n^2+428*n-376)*x^2+(8*n^3-72*n^2+192*n-160)*x^3)*P(n-2,x) + (n^3-6*n^2+13*n-8+(6*n^3-42*n^2+96*n-68)*x+(12*n^3-96*n^2+236*n-184)*x^2+(8*n^3-72*n^2+192*n-160)*x^3)*d/dx P(n-2,x) + (7*n^2-19*n+12+(12*n^2-40*n+28)*x-(4*n^2-28*n+40)*x^2)*P(n-1,x) - (n^2-5*n+8+(4*n^2-24*n+36)*x+(4*n^2-28*n+40)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 4 + 2*x, P(3,x) = 15 + 18*x + 4*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[3, x_] := 15 + 18*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(5*n^3 - 44*n^2 + 139*n - 156 + (32*n^3 - 308*n^2 + 992*n - 1068)*x + (72*n^3 - 744*n^2 + 2424*n - 2520)*x^2 + (64*n^3 - 688*n^2 + 2240*n - 2256)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (8*n^3 - 68*n^2 + 200*n - 204)*x + (24*n^3 - 216*n^2 + 648*n - 648)*x^2 + (32*n^3 - 304*n^2 + 928*n - 912)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*D[P[n - 3, x], x] + (n^3 - 18*n^2 + 65*n - 68 + (6*n^3 - 90*n^2 + 304*n - 292)*x + (12*n^3 - 144*n^2 + 428*n - 376)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (6*n^3 - 42*n^2 + 96*n - 68)*x + (12*n^3 - 96*n^2 + 236*n - 184)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*D[P[n - 2, x], x] + (7*n^2 - 19*n + 12 + (12*n^2 - 40*n + 28)*x - (4*n^2 - 28*n + 40)*x^2)*P[n - 1, x] - (n^2 - 5*n + 8 + (4*n^2 - 24*n + 36)*x + (4*n^2 - 28*n + 40)*x^2)*D[P[n - 1, x], x])/(4*n - 8)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; v[3] = 15 + 18*x + 4*x^2; for(n = 4, N, my(i = n); v[i] = (-(5*n^3 - 44*n^2 + 139*n - 156 + (32*n^3 - 308*n^2 + 992*n - 1068)*x + (72*n^3 - 744*n^2 + 2424*n - 2520)*x^2 + (64*n^3 - 688*n^2 + 2240*n - 2256)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (8*n^3 - 68*n^2 + 200*n - 204)*x + (24*n^3 - 216*n^2 + 648*n - 648)*x^2 + (32*n^3 - 304*n^2 + 928*n - 912)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*deriv(v[i - 3], x) + (n^3 - 18*n^2 + 65*n - 68 + (6*n^3 - 90*n^2 + 304*n - 292)*x + (12*n^3 - 144*n^2 + 428*n - 376)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (6*n^3 - 42*n^2 + 96*n - 68)*x + (12*n^3 - 96*n^2 + 236*n - 184)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*deriv(v[i - 2], x) + (7*n^2 - 19*n + 12 + (12*n^2 - 40*n + 28)*x - (4*n^2 - 28*n + 40)*x^2)*v[i - 1] - (n^2 - 5*n + 8 + (4*n^2 - 24*n + 36)*x + (4*n^2 - 28*n + 40)*x^2)*deriv(v[i - 1], x))/(4*n - 8)); 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_A013609 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