A046716 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-12 + 6\,n\right)\,P_{n}(x) = \left(222 + 2250\,x + 7776\,x^{2} + 9882\,x^{3} + 2430\,x^{4} - 197\,n - 2082\,n x - 7452\,n x^{2} - 9774\,n x^{3} - 2511\,n x^{4} + 62\,n^{2} + 642\,n^{2} x + 2268\,n^{2} x^{2} + 2970\,n^{2} x^{3} + 810\,n^{2} x^{4} - 7\,n^{3} - 66\,n^{3} x - 216\,n^{3} x^{2} - 270\,n^{3} x^{3} - 81\,n^{3} x^{4}\right)\,P_{n-3}(x) + \left(24 + 306\,x + 1458\,x^{2} + 3078\,x^{3} + 2430\,x^{4} - 23\,n - 300\,n x - 1458\,n x^{2} - 3132\,n x^{3} - 2511\,n x^{4} + 8\,n^{2} + 102\,n^{2} x + 486\,n^{2} x^{2} + 1026\,n^{2} x^{3} + 810\,n^{2} x^{4} - n^{3} - 12\,n^{3} x - 54\,n^{3} x^{2} - 108\,n^{3} x^{3} - 81\,n^{3} x^{4}\right)\,P_{n-3}'(x) + \left(-98 - 606\,x - 1062\,x^{2} - 540\,x^{3} + 91\,n + 612\,n x + 1179\,n x^{2} + 648\,n x^{3} - 24\,n^{2} - 171\,n^{2} x - 378\,n^{2} x^{2} - 243\,n^{2} x^{3} + n^{3} + 9\,n^{3} x + 27\,n^{3} x^{2} + 27\,n^{3} x^{3}\right)\,P_{n-2}(x) + \left(-8 - 102\,x - 414\,x^{2} - 540\,x^{3} + 13\,n + 144\,n x + 531\,n x^{2} + 648\,n x^{3} - 6\,n^{2} - 63\,n^{2} x - 216\,n^{2} x^{2} - 243\,n^{2} x^{3} + n^{3} + 9\,n^{3} x + 27\,n^{3} x^{2} + 27\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(22 + 90\,x - 90\,x^{2} - 31\,n - 108\,n x + 63\,n x^{2} + 11\,n^{2} + 30\,n^{2} x - 9\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-8 - 54\,x - 90\,x^{2} + 5\,n + 36\,n x + 63\,n x^{2} - n^{2} - 6\,n^{2} x - 9\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 5 + 3\,x\)

\(P_{3}(x) = 20 + 33\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	5	3							8
3	20	33	9						62
4	91	273	180	27					571
5	497	2358	3231	1080	81				7247
6	3209	21855	50877	44253	8829	243			129266
7	23896	217131	743247	1156113	746739	106677	729		2994532
8	201275	2313933	10637055	24578208	29034369	15143274	1749600	2187	83659901

All 15 rows

Technical details

Node id: n-4994171eacab0188 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d6534fa70ef437bb

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:        5,         3;
   3:       20,        33,          9;
   4:       91,       273,        180,         27;
   5:      497,      2358,       3231,       1080,          81;
   6:     3209,     21855,      50877,      44253,        8829,         243;
   7:    23896,    217131,     743247,    1156113,      746739,      106677,         729;
   8:   201275,   2313933,   10637055,   24578208,    29034369,    15143274,     1749600,       2187;
   9:  1890809,  26466228,  154093158,  478759572,   841902579,   807935148,   356357070,   35219448,      6561;
  10: 19599517, 324657675, 2300067090, 9043779546, 21364136451, 30492700209, 24756106626, 9499186386, 824776749, 19683;

%F Formula

The row polynomials satisfy (6*n-12)*P(n,x) = -(7*n^3-62*n^2+197*n-222+(66*n^3-642*n^2+2082*n-2250)*x+(216*n^3-2268*n^2+7452*n-7776)*x^2+(270*n^3-2970*n^2+9774*n-9882)*x^3+(81*n^3-810*n^2+2511*n-2430)*x^4)*P(n-3,x) - (n^3-8*n^2+23*n-24+(12*n^3-102*n^2+300*n-306)*x+(54*n^3-486*n^2+1458*n-1458)*x^2+(108*n^3-1026*n^2+3132*n-3078)*x^3+(81*n^3-810*n^2+2511*n-2430)*x^4)*d/dx P(n-3,x) + (n^3-24*n^2+91*n-98+(9*n^3-171*n^2+612*n-606)*x+(27*n^3-378*n^2+1179*n-1062)*x^2+(27*n^3-243*n^2+648*n-540)*x^3)*P(n-2,x) + (n^3-6*n^2+13*n-8+(9*n^3-63*n^2+144*n-102)*x+(27*n^3-216*n^2+531*n-414)*x^2+(27*n^3-243*n^2+648*n-540)*x^3)*d/dx P(n-2,x) + (11*n^2-31*n+22+(30*n^2-108*n+90)*x-(9*n^2-63*n+90)*x^2)*P(n-1,x) - (n^2-5*n+8+(6*n^2-36*n+54)*x+(9*n^2-63*n+90)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 5 + 3*x, P(3,x) = 20 + 33*x + 9*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 5 + 3*x;
P[3, x_] := 20 + 33*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(7*n^3 - 62*n^2 + 197*n - 222 + (66*n^3 - 642*n^2 + 2082*n - 2250)*x + (216*n^3 - 2268*n^2 + 7452*n - 7776)*x^2 + (270*n^3 - 2970*n^2 + 9774*n - 9882)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (12*n^3 - 102*n^2 + 300*n - 306)*x + (54*n^3 - 486*n^2 + 1458*n - 1458)*x^2 + (108*n^3 - 1026*n^2 + 3132*n - 3078)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*D[P[n - 3, x], x] + (n^3 - 24*n^2 + 91*n - 98 + (9*n^3 - 171*n^2 + 612*n - 606)*x + (27*n^3 - 378*n^2 + 1179*n - 1062)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (9*n^3 - 63*n^2 + 144*n - 102)*x + (27*n^3 - 216*n^2 + 531*n - 414)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*D[P[n - 2, x], x] + (11*n^2 - 31*n + 22 + (30*n^2 - 108*n + 90)*x - (9*n^2 - 63*n + 90)*x^2)*P[n - 1, x] - (n^2 - 5*n + 8 + (6*n^2 - 36*n + 54)*x + (9*n^2 - 63*n + 90)*x^2)*D[P[n - 1, x], x])/(6*n - 12)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 3*x; v[3] = 20 + 33*x + 9*x^2; for(n = 4, N, my(i = n); v[i] = (-(7*n^3 - 62*n^2 + 197*n - 222 + (66*n^3 - 642*n^2 + 2082*n - 2250)*x + (216*n^3 - 2268*n^2 + 7452*n - 7776)*x^2 + (270*n^3 - 2970*n^2 + 9774*n - 9882)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (12*n^3 - 102*n^2 + 300*n - 306)*x + (54*n^3 - 486*n^2 + 1458*n - 1458)*x^2 + (108*n^3 - 1026*n^2 + 3132*n - 3078)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*deriv(v[i - 3], x) + (n^3 - 24*n^2 + 91*n - 98 + (9*n^3 - 171*n^2 + 612*n - 606)*x + (27*n^3 - 378*n^2 + 1179*n - 1062)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (9*n^3 - 63*n^2 + 144*n - 102)*x + (27*n^3 - 216*n^2 + 531*n - 414)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*deriv(v[i - 2], x) + (11*n^2 - 31*n + 22 + (30*n^2 - 108*n + 90)*x - (9*n^2 - 63*n + 90)*x^2)*v[i - 1] - (n^2 - 5*n + 8 + (6*n^2 - 36*n + 54)*x + (9*n^2 - 63*n + 90)*x^2)*deriv(v[i - 1], x))/(6*n - 12)); 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_A013610 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