A257606 → D+1

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + n\right)\,P_{n}(x) = \left(-40 - 48\,x - 16\,x^{2} - 12\,x^{3} - 4\,x^{4} - 20\,n - 20\,n x - 24\,n x^{2} - 8\,n x^{3} - 2\,n x^{4} - 8\,n^{2} x - 4\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-8\,x - 2\,x^{2} + 6\,x^{3} + 2\,x^{4} + 2\,x^{5} - 4\,n x + n x^{2} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(13 + 21\,x + 6\,x^{2} + 2\,x^{3} + 9\,n + 5\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(x + x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 8 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	8	4							12
2	56	72	16						144
3	360	888	488	64					1800
4	2184	9032	9336	2952	256				23760
5	12712	81304	139232	81560	16808	1024			332640
6	71752	672360	1764288	1696720	640632	92232	4096		4942080
7	395624	5225272	19913016	29150000	17945080	4698456	493928	16384	77837760

All 15 rows

Technical details

Node id: n-b972da6d709613ba · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-23daeb7237195c22

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:        8,         4;
  2:       56,        72,         16;
  3:      360,       888,        488,         64;
  4:     2184,      9032,       9336,       2952,        256;
  5:    12712,     81304,     139232,      81560,      16808,       1024;
  6:    71752,    672360,    1764288,    1696720,     640632,      92232,       4096;
  7:   395624,   5225272,   19913016,   29150000,   17945080,    4698456,     493928,     16384;
  8:  2141960,  38737416,  206181928,  435737144,  406365720,  172600120,   32865464,   2600712,    65536;
  9: 11430696, 276815832, 1997698544, 5860802152, 7858765776, 5039544104, 1551390448, 222172056, 13527848, 262144;

%F Formula

The row polynomials satisfy (n+1+2*x)*P(n,x) = -(20*n+40+(8*n^2+20*n+48)*x+(n^3+4*n^2+24*n+16)*x^2+(3*n^2+8*n+12)*x^3+(2*n+4)*x^4)*P(n-2,x) - ((4*n+8)*x+(n^2-n+2)*x^2-(n^2+6)*x^3-(3*n+2)*x^4-2*x^5)*d/dx P(n-2,x) + (9*n+13+(2*n^2+5*n+21)*x+(5*n+6)*x^2+2*x^3)*P(n-1,x) + ((n+1)*x-(n-1)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 8 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(20*n + 40 + (8*n^2 + 20*n + 48)*x + (n^3 + 4*n^2 + 24*n + 16)*x^2 + (3*n^2 + 8*n + 12)*x^3 + (2*n + 4)*x^4)*P[n - 2, x] - ((4*n + 8)*x + (n^2 - n + 2)*x^2 - (n^2 + 6)*x^3 - (3*n + 2)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (9*n + 13 + (2*n^2 + 5*n + 21)*x + (5*n + 6)*x^2 + 2*x^3)*P[n - 1, x] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n + 1 + 2*x)]];
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] = 8 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(20*n + 40 + (8*n^2 + 20*n + 48)*x + (n^3 + 4*n^2 + 24*n + 16)*x^2 + (3*n^2 + 8*n + 12)*x^3 + (2*n + 4)*x^4)*v[i - 2] - ((4*n + 8)*x + (n^2 - n + 2)*x^2 - (n^2 + 6)*x^3 - (3*n + 2)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (9*n + 13 + (2*n^2 + 5*n + 21)*x + (5*n + 6)*x^2 + 2*x^3)*v[i - 1] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n + 1 + 2*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4.?
    
    ↑ Liu–Wang theorem (above)

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