A008957 → D

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-n^{2} - 8\,n^{2} x - 4\,n^{2} x^{2} + 8\,n^{3} x + 12\,n^{3} x^{2} - 2\,n^{4} x - 13\,n^{4} x^{2} + 6\,n^{5} x^{2} - n^{6} x^{2}\right)\,P_{n-2}(x) + \left(-6\,n^{2} x^{2} - 5\,n^{2} x^{3} + 2\,n^{3} x^{2} + 12\,n^{3} x^{3} - 9\,n^{4} x^{3} + 2\,n^{5} x^{3}\right)\,P_{n-2}'(x) + \left(-n^{2} x^{3} - n^{2} x^{4} + 2\,n^{3} x^{4} - n^{4} x^{4}\right)\,P_{n-2}''(x) + \left(1 + 2\,x - 2\,n - 8\,n x + 2\,n^{2} + 12\,n^{2} x - 8\,n^{3} x + 2\,n^{4} x\right)\,P_{n-1}(x) + \left(4\,x^{2} - 10\,n x^{2} + 8\,n^{2} x^{2} - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x) + \left(x^{3} - 2\,n x^{3} + n^{2} x^{3}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	5	2						7
3	14	42	3					59
4	30	294	255	4				583
5	55	1254	4224	1364	5			6902
6	91	4004	34320	52052	6825	6		97298
7	140	10556	184470	784212	592410	32766	7	1604561

All 15 rows

Technical details

Node id: n-ee02330825df0350 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a07a15ae93e03f31

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   0;
  1:   1;
  2:   5,     2;
  3:  14,    42,       3;
  4:  30,   294,     255,        4;
  5:  55,  1254,    4224,     1364,         5;
  6:  91,  4004,   34320,    52052,      6825,         6;
  7: 140, 10556,  184470,   784212,    592410,     32766,        7;
  8: 204, 24276,  754494,  6933212,  16276650,   6430794,   152915,      8;
  9: 285, 50388, 2538780, 43148924, 232939525, 318942474, 67676252, 699048, 9;

%F Formula

The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^2+(2*n^4-8*n^3+8*n^2)*x+(n^6-6*n^5+13*n^4-12*n^3+4*n^2)*x^2)*P(n-2,x) + ((2*n^3-6*n^2)*x^2+(2*n^5-9*n^4+12*n^3-5*n^2)*x^3)*d/dx P(n-2,x) - (n^2*x^3+(n^4-2*n^3+n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2-2*n+1+(2*n^4-8*n^3+12*n^2-8*n+2)*x)*P(n-1,x) - (2*n^3-8*n^2+10*n-4)*x^2*d/dx P(n-1,x) + (n^2-2*n+1)*x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + (2*n^4 - 8*n^3 + 8*n^2)*x + (n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^2)*x^2)*P[n - 2, x] + ((2*n^3 - 6*n^2)*x^2 + (2*n^5 - 9*n^4 + 12*n^3 - 5*n^2)*x^3)*D[P[n - 2, x], x] - (n^2*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 2*n + 1 + (2*n^4 - 8*n^3 + 12*n^2 - 8*n + 2)*x)*P[n - 1, x] - (2*n^3 - 8*n^2 + 10*n - 4)*x^2*D[P[n - 1, x], x] + (n^2 - 2*n + 1)*x^3*D[P[n - 1, x], {x, 2}])/(n^2 - 2*n + 1)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + (2*n^4 - 8*n^3 + 8*n^2)*x + (n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^2)*x^2)*v[i - 2] + ((2*n^3 - 6*n^2)*x^2 + (2*n^5 - 9*n^4 + 12*n^3 - 5*n^2)*x^3)*deriv(v[i - 2], x) - (n^2*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 2*n + 1 + (2*n^4 - 8*n^3 + 12*n^2 - 8*n + 2)*x)*v[i - 1] - (2*n^3 - 8*n^2 + 10*n - 4)*x^2*deriv(v[i - 1], x) + (n^2 - 2*n + 1)*x^3*deriv(deriv(v[i - 1], x), x))/(n^2 - 2*n + 1)); 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 Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.