A257626 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,x + n\right)\,P_{n}(x) = \left(-108\,x + 108\,x^{2} - 36\,x^{3} - 54\,n + 54\,n x - 90\,n x^{2} + 36\,n x^{3} - 18\,n x^{4} - 36\,n^{2} x + 18\,n^{2} x^{2} - 27\,n^{2} x^{3} - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-36\,x^{2} + 54\,x^{3} - 36\,x^{4} + 18\,x^{5} - 18\,n x + 27\,n x^{2} - 36\,n x^{3} + 27\,n x^{4} - 9\,n^{2} x^{2} + 9\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-9 + 33\,x - 12\,x^{2} + 6\,x^{3} + 15\,n - 9\,n x + 15\,n x^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x + 9\,x^{2} - 6\,x^{3} + 3\,n x - 3\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 12 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	12	6							18
2	144	180	36						360
3	1620	4212	2052	216					8100
4	17172	83268	81324	21060	1296				204120
5	173988	1451196	2569968	1307340	205092	7776			5715360
6	1705860	23062644	68526000	62081640	18997740	1939140	46656		176359680
7	16332516	342580428	1608521004	2408761800	1298071980	259579404	18012132	279936	5952139200

All 15 rows

Technical details

Node id: n-4969942ab2f8f858 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-9f7813ee7004ecb1

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 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:         12,           6;
  2:        144,         180,           36;
  3:       1620,        4212,         2052,           216;
  4:      17172,       83268,        81324,         21060,          1296;
  5:     173988,     1451196,      2569968,       1307340,        205092,          7776;
  6:    1705860,    23062644,     68526000,      62081640,      18997740,       1939140,        46656;
  7:   16332516,   342580428,   1608521004,    2408761800,    1298071980,     259579404,     18012132,      279936;
  8:  153711108,  4840889508,  34308912420,   80027167644,   70617223980,   24721401852,   3408649452,   165468420,    1679616;
  9: 1428749604, 65894668668, 680504138136, 2359796365908, 3215561478984, 1832459192676, 441754701912, 43587373644, 1509371172, 10077696;

%F Formula

The row polynomials satisfy (n-1+2*x)*P(n,x) = -(54*n+(36*n^2-54*n+108)*x+(9*n^3-18*n^2+90*n-108)*x^2+(27*n^2-36*n+36)*x^3+18*n*x^4)*P(n-2,x) - (18*n*x+(9*n^2-27*n+36)*x^2-(9*n^2-36*n+54)*x^3-(27*n-36)*x^4-18*x^5)*d/dx P(n-2,x) + (15*n-9+(6*n^2-9*n+33)*x+(15*n-12)*x^2+6*x^3)*P(n-1,x) + ((3*n-3)*x-(3*n-9)*x^2-6*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 12 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 12 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(54*n + (36*n^2 - 54*n + 108)*x + (9*n^3 - 18*n^2 + 90*n - 108)*x^2 + (27*n^2 - 36*n + 36)*x^3 + 18*n*x^4)*P[n - 2, x] - (18*n*x + (9*n^2 - 27*n + 36)*x^2 - (9*n^2 - 36*n + 54)*x^3 - (27*n - 36)*x^4 - 18*x^5)*D[P[n - 2, x], x] + (15*n - 9 + (6*n^2 - 9*n + 33)*x + (15*n - 12)*x^2 + 6*x^3)*P[n - 1, x] + ((3*n - 3)*x - (3*n - 9)*x^2 - 6*x^3)*D[P[n - 1, x], x])/(n - 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 12 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(54*n + (36*n^2 - 54*n + 108)*x + (9*n^3 - 18*n^2 + 90*n - 108)*x^2 + (27*n^2 - 36*n + 36)*x^3 + 18*n*x^4)*v[i - 2] - (18*n*x + (9*n^2 - 27*n + 36)*x^2 - (9*n^2 - 36*n + 54)*x^3 - (27*n - 36)*x^4 - 18*x^5)*deriv(v[i - 2], x) + (15*n - 9 + (6*n^2 - 9*n + 33)*x + (15*n - 12)*x^2 + 6*x^3)*v[i - 1] + ((3*n - 3)*x - (3*n - 9)*x^2 - 6*x^3)*deriv(v[i - 1], x))/(n - 1 + 2*x)); 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 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.