A059344 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-15 - 198\,x + 108\,x^{2} - 45\,n + 108\,n x + 27\,n^{2}\right)\,P_{n}(x) = \left(-3 + 180\,x + 432\,x^{2} - 648\,x^{3} + 33\,n - 216\,n x - 972\,n x^{2} + 648\,n x^{3} - 117\,n^{2} - 216\,n^{2} x + 648\,n^{2} x^{2} + 54\,n^{3} + 162\,n^{3} x\right)\,P_{n-1}(x) + \left(-4 - 48\,x - 324\,x^{2} - 1080\,x^{3} - 1296\,x^{4} - 12\,n - 216\,n x - 972\,n x^{2} - 1296\,n x^{3} - 36\,n^{2} - 216\,n^{2} x - 324\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	9	6							15
3	25	18							43
4	133	324	108						565
5	501	1500	540						2541
6	2581	11970	14580	3240					32371
7	11509	63126	94500	22680					191815
8	61545	433608	1005480	816480	136080				2453193
9	308233	2485944	6817608	6804000	1224720				17640505
10	1736641	16617150	58537080	90493200	55112400	7348320			229844791
11	9508401	101716890	410180760	749936880	561330000	80831520			1913504451
12	56535469	687709836	3290195700	7726894560	8958826800	4364902080	484989120		25570053565
13	332930365	4449931668	23801752260	63988198560	87742614960	52540488000	6304858560		239160774373
14	2084407053	30868366074	187744785228	598815617400	1054721107440	978303886560	397206089280	37829151360	3287573410395

Technical details

Node id: n-42ad00402123d391 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-4fd8c88feba28db3

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:      9,       6;
  3:     25,      18;
  4:    133,     324,     108;
  5:    501,    1500,     540;
  6:   2581,   11970,   14580,    3240;
  7:  11509,   63126,   94500,   22680;
  8:  61545,  433608, 1005480,  816480,  136080;
  9: 308233, 2485944, 6817608, 6804000, 1224720;

%F Formula

The row polynomials satisfy (27*n^2-45*n-15+(108*n-198)*x+108*x^2)*P(n,x) = (54*n^3-117*n^2+33*n-3+(162*n^3-216*n^2-216*n+180)*x+(648*n^2-972*n+432)*x^2+(648*n-648)*x^3)*P(n-1,x) - (36*n^2+12*n+4+(216*n^2+216*n+48)*x+(324*n^2+972*n+324)*x^2+(1296*n+1080)*x^3+1296*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((54*n^3 - 117*n^2 + 33*n - 3 + (162*n^3 - 216*n^2 - 216*n + 180)*x + (648*n^2 - 972*n + 432)*x^2 + (648*n - 648)*x^3)*P[n - 1, x] - (36*n^2 + 12*n + 4 + (216*n^2 + 216*n + 48)*x + (324*n^2 + 972*n + 324)*x^2 + (1296*n + 1080)*x^3 + 1296*x^4)*D[P[n - 1, x], x])/(27*n^2 - 45*n - 15 + (108*n - 198)*x + 108*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((54*n^3 - 117*n^2 + 33*n - 3 + (162*n^3 - 216*n^2 - 216*n + 180)*x + (648*n^2 - 972*n + 432)*x^2 + (648*n - 648)*x^3)*v[i - 1] - (36*n^2 + 12*n + 4 + (216*n^2 + 216*n + 48)*x + (324*n^2 + 972*n + 324)*x^2 + (1296*n + 1080)*x^3 + 1296*x^4)*deriv(v[i - 1], x))/(27*n^2 - 45*n - 15 + (108*n - 198)*x + 108*x^2)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013610 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)

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.