A151775 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 + 4\,x + 22\,x^{2} + 16\,x^{3} - 8\,x^{4}\right)\,P_{n-2}(x) + \left(6 + 40\,x + 138\,x^{2} + 156\,x^{3} + 12\,x^{4} - 40\,x^{5}\right)\,P_{n-2}'(x) + \left(4\,x + 32\,x^{2} + 60\,x^{3} + 24\,x^{4} - 24\,x^{5} - 16\,x^{6}\right)\,P_{n-2}''(x) + \left(8 - 14\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x) + \left(6 + 28\,x + 22\,x^{2}\right)\,P_{n-1}'(x) + \left(4\,x + 8\,x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	33	76	24						133
3	723	3302	3480	720					8225
4	25953	191896	386064	262080	40320				906313
5	1376643	14824202	46756560	59300640	29635200	3628800			155522045
6	101031873	1483727716	6631273704	12938016960	11920487040	4710182400	479001600		38263721293
7	9795436563	187528805102	1116344796840	3035876603760	4251296649600	3069402336000	1002550348800	87178291200	12759973267865

All 15 rows

Technical details

Node id: n-fe550175f722890a · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-ad67aec3cd3f88df

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:               3,                2;
  2:              33,               76,                24;
  3:             723,             3302,              3480,                720;
  4:           25953,           191896,            386064,             262080,              40320;
  5:         1376643,         14824202,          46756560,           59300640,           29635200,            3628800;
  6:       101031873,       1483727716,        6631273704,        12938016960,        11920487040,         4710182400,          479001600;
  7:      9795436563,     187528805102,     1116344796840,      3035876603760,      4251296649600,      3069402336000,      1002550348800,        87178291200;
  8:   1212135593793,   29265860447536,   222494217046944,    794354143040640,   1533871279100160,   1669219346995200,    989822318284800,    275483400192000,    20922789888000;
  9: 186388033956483, 5532759960522002, 52101046426636320, 234537391905814080, 588855455629459200, 876964309110489600, 779061750321254400, 391736348933529600, 94968543301632000, 6402373705728000;

%F Formula

The row polynomials satisfy P(n,x) = -(3-4*x-22*x^2-16*x^3+8*x^4)*P(n-2,x) + (6+40*x+138*x^2+156*x^3+12*x^4-40*x^5)*d/dx P(n-2,x) + (4*x+32*x^2+60*x^3+24*x^4-24*x^5-16*x^6)*d^2/dx^2 P(n-2,x) + (8-14*x^2+4*x^3)*P(n-1,x) + (6+28*x+22*x^2)*d/dx P(n-1,x) + (4*x+8*x^2+4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(3 - 4*x - 22*x^2 - 16*x^3 + 8*x^4)*P[n - 2, x] + (6 + 40*x + 138*x^2 + 156*x^3 + 12*x^4 - 40*x^5)*D[P[n - 2, x], x] + (4*x + 32*x^2 + 60*x^3 + 24*x^4 - 24*x^5 - 16*x^6)*D[P[n - 2, x], {x, 2}] + (8 - 14*x^2 + 4*x^3)*P[n - 1, x] + (6 + 28*x + 22*x^2)*D[P[n - 1, x], x] + (4*x + 8*x^2 + 4*x^3)*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] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 - 4*x - 22*x^2 - 16*x^3 + 8*x^4)*v[i - 2] + (6 + 40*x + 138*x^2 + 156*x^3 + 12*x^4 - 40*x^5)*deriv(v[i - 2], x) + (4*x + 32*x^2 + 60*x^3 + 24*x^4 - 24*x^5 - 16*x^6)*deriv(deriv(v[i - 2], x), x) + (8 - 14*x^2 + 4*x^3)*v[i - 1] + (6 + 28*x + 22*x^2)*deriv(v[i - 1], x) + (4*x + 8*x^2 + 4*x^3)*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) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.?
  - 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.