A036969 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	5	2						7
3	21	28	3					52
4	85	294	90	4				473
5	341	2816	1881	220	5			5263
6	1365	26026	34320	8008	455	6		70180
7	5461	236964	588159	245960	26390	840	7	1103781

All 15 rows

Technical details

Node id: n-a4a84eec33e1a48f · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-381b673846e6e94e

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:    21,       28,         3;
  4:    85,      294,        90,         4;
  5:   341,     2816,      1881,       220,        5;
  6:  1365,    26026,     34320,      8008,      455,       6;
  7:  5461,   236964,    588159,    245960,    26390,     840,      7;
  8: 21845,  2143598,   9765990,   6933212,  1257490,   72828,   1428,    8;
  9: 87381, 19336072, 159471237, 186351620, 53936155, 5077560, 176358, 2280, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(4+2*x+x^2)*P(n-2,x) - (5*x+4*x^2)*d/dx P(n-2,x) - (x^2+x^3)*d^2/dx^2 P(n-2,x) + (5+2*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) + x^2*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[-(4 + 2*x + x^2)*P[n - 2, x] - (5*x + 4*x^2)*D[P[n - 2, x], x] - (x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (5 + 2*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {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] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(4 + 2*x + x^2)*v[i - 2] - (5*x + 4*x^2)*deriv(v[i - 2], x) - (x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (5 + 2*x)*v[i - 1] + 5*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] 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 [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.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - ↑ Iterated real-rootedness preservers (above)

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.