A152842 → D

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(5 + 21\,x + 18\,x^{2} - 9\,n - 33\,n x - 27\,n x^{2} + 4\,n^{2} + 12\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 18\,x + 54\,x^{2} + 54\,x^{3} - 23\,n - 108\,n x - 138\,n x^{2} + 27\,n x^{3} + 108\,n x^{4} + 12\,n^{2} + 20\,n^{2} x - 87\,n^{2} x^{2} - 243\,n^{2} x^{3} - 162\,n^{2} x^{4} + 16\,n^{3} x + 72\,n^{3} x^{2} + 108\,n^{3} x^{3} + 54\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(1 + 10\,x + 36\,x^{2} + 54\,x^{3} + 27\,x^{4} - 4\,n - 26\,n x - 46\,n x^{2} + 12\,n x^{3} + 90\,n x^{4} + 54\,n x^{5} - 8\,n^{2} x - 56\,n^{2} x^{2} - 138\,n^{2} x^{3} - 144\,n^{2} x^{4} - 54\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	3							3
2	4	6						10
3	7	30	27					64
4	8	44	72	36				160
5	11	92	270	324	135			832
6	12	114	408	684	540	162		1920
7	15	186	921	2316	3105	2106	567	9216

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-0eece95308901031 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-411c25b8cf1f8d73

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): yes

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  0;
  1:  3;
  2:  4,   6;
  3:  7,  30,   27;
  4:  8,  44,   72,   36;
  5: 11,  92,  270,  324,   135;
  6: 12, 114,  408,  684,   540,   162;
  7: 15, 186,  921, 2316,  3105,  2106,   567;
  8: 16, 216, 1200, 3544,  6000,  5832,  3024,   648;
  9: 19, 312, 2172, 8344, 19290, 27432, 23436, 11016, 2187;

%F Formula

The row polynomials satisfy (4*n^2-9*n+5+(12*n^2-33*n+21)*x+(9*n^2-27*n+18)*x^2)*P(n,x) = (12*n^2-23*n+2+(16*n^3+20*n^2-108*n+18)*x+(72*n^3-87*n^2-138*n+54)*x^2+(108*n^3-243*n^2+27*n+54)*x^3+(54*n^3-162*n^2+108*n)*x^4)*P(n-1,x) - (4*n-1+(8*n^2+26*n-10)*x+(56*n^2+46*n-36)*x^2+(138*n^2-12*n-54)*x^3+(144*n^2-90*n-27)*x^4+(54*n^2-54*n)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((12*n^2 - 23*n + 2 + (16*n^3 + 20*n^2 - 108*n + 18)*x + (72*n^3 - 87*n^2 - 138*n + 54)*x^2 + (108*n^3 - 243*n^2 + 27*n + 54)*x^3 + (54*n^3 - 162*n^2 + 108*n)*x^4)*P[n - 1, x] - (4*n - 1 + (8*n^2 + 26*n - 10)*x + (56*n^2 + 46*n - 36)*x^2 + (138*n^2 - 12*n - 54)*x^3 + (144*n^2 - 90*n - 27)*x^4 + (54*n^2 - 54*n)*x^5)*D[P[n - 1, x], x])/(4*n^2 - 9*n + 5 + (12*n^2 - 33*n + 21)*x + (9*n^2 - 27*n + 18)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((12*n^2 - 23*n + 2 + (16*n^3 + 20*n^2 - 108*n + 18)*x + (72*n^3 - 87*n^2 - 138*n + 54)*x^2 + (108*n^3 - 243*n^2 + 27*n + 54)*x^3 + (54*n^3 - 162*n^2 + 108*n)*x^4)*v[i - 1] - (4*n - 1 + (8*n^2 + 26*n - 10)*x + (56*n^2 + 46*n - 36)*x^2 + (138*n^2 - 12*n - 54)*x^3 + (144*n^2 - 90*n - 27)*x^4 + (54*n^2 - 54*n)*x^5)*deriv(v[i - 1], x))/(4*n^2 - 9*n + 5 + (12*n^2 - 33*n + 21)*x + (9*n^2 - 27*n + 18)*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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-6c646dd86721ff4d
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-6c646dd86721ff4d
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-6c646dd86721ff4d

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image derivative_polynomial, shift 0 e-6c646dd86721ff4d

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-596b986e3a00db44 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

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