A052553 → D → D

nonnegative coefficients

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

Recurrence

\(\left(-5 + n\right)\,P_{n}(x) = \left(-3 - 12\,x + n + 2\,n x\right)\,P_{n-1}(x) + \left(-x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 0\)

\(P_{3}(x) = 0\)

\(P_{4}(x) = 0\)

\(P_{5}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	P_n(1)
1							0
2							0
3							0
4							0
5	2						2
6	6						6
7	12	6					18
8	20	24					44
9	30	60	12				102
10	42	120	60				222
11	56	210	180	20			466
12	72	336	420	120			948
13	90	504	840	420	30		1884
14	110	720	1512	1120	210		3672
15	132	990	2520	2520	840	42	7044

Technical details

Node id: n-f4987202b5bfbff2 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-ddd039a18eb93f86

Observed on the stored rows ?

Degrees: -1, -1, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5 (half affine: slope 1, intercept -5)
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:  0;
   2:  0;
   3:  0;
   4:  0;
   5:  2;
   6:  6;
   7: 12,   6;
   8: 20,  24;
   9: 30,  60, 12;
  10: 42, 120, 60;

%F Formula

The row polynomials satisfy (n-5)*P(n,x) = (n-3+(2*n-12)*x)*P(n-1,x) - (x+4*x^2)*d/dx P(n-1,x) for n >= 6, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 0, P(4,x) = 0, P(5,x) = 2.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 0;
P[4, x_] := 0;
P[5, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (2*n - 12)*x)*P[n - 1, x] - (x + 4*x^2)*D[P[n - 1, x], x])/(n - 5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 0; v[4] = 0; v[5] = 2; for(n = 6, N, my(i = n); v[i] = ((n - 3 + (2*n - 12)*x)*v[i - 1] - (x + 4*x^2)*deriv(v[i - 1], x))/(n - 5)); 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 derivative of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonal…?
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.?
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
    no recorded property covers this requirement
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
    no recorded property covers this requirement
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from delannoy?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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 delannoy?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from delannoy?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
    no recorded property covers this requirement
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
    no recorded property covers this requirement
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (above)
    
    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
    
    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 basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.?
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
    
    Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
    no recorded property covers this requirement
    
    ↑ Ma–Wang interval theorem (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Geometric basis map (elementary) (above)
    
    ↑ Geometric basis map (elementary) (above)
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Ma–Wang interval theorem (above)
    
    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
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.?
    
    ↑ Gamma criterion (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.