A257613 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,x + n\right)\,P_{n}(x) = \left(-48\,x + 48\,x^{2} - 16\,x^{3} - 24\,n + 24\,n x - 40\,n x^{2} + 16\,n x^{3} - 8\,n x^{4} - 16\,n^{2} x + 8\,n^{2} x^{2} - 12\,n^{2} x^{3} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-16\,x^{2} + 24\,x^{3} - 16\,x^{4} + 8\,x^{5} - 8\,n x + 12\,n x^{2} - 16\,n x^{3} + 12\,n x^{4} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-6 + 22\,x - 8\,x^{2} + 4\,x^{3} + 10\,n - 6\,n x + 10\,n x^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 6\,x^{2} - 4\,x^{3} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	8	4							12
2	64	80	16						160
3	480	1248	608	64					2400
4	3392	16448	16064	4160	256				40320
5	22912	191104	338432	172160	27008	1024			752640
6	149760	2024704	6016000	5450240	1667840	170240	4096		15482880
7	955904	20050432	94142976	140979200	75973120	15192576	1054208	16384	348364800

All 15 rows

Technical details

Node id: n-fa12f2f21ad6abc1 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-9f7813ee7004ecb1

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:       64,         80,          16;
  3:      480,       1248,         608,          64;
  4:     3392,      16448,       16064,        4160,         256;
  5:    22912,     191104,      338432,      172160,       27008,        1024;
  6:   149760,    2024704,     6016000,     5450240,     1667840,      170240,        4096;
  7:   955904,   20050432,    94142976,   140979200,    75973120,    15192576,     1054208,      16384;
  8:  5997568,  188883968,  1338680320,  3122535424,  2755374080,   964590592,   133000192,    6456320,    65536;
  9: 37165056, 1714071552, 17701474304, 61383718912, 83644133376, 47666468864, 11491053568, 1133807616, 39262208, 262144;

%F Formula

The row polynomials satisfy (n-1+2*x)*P(n,x) = -(24*n+(16*n^2-24*n+48)*x+(4*n^3-8*n^2+40*n-48)*x^2+(12*n^2-16*n+16)*x^3+8*n*x^4)*P(n-2,x) - (8*n*x+(4*n^2-12*n+16)*x^2-(4*n^2-16*n+24)*x^3-(12*n-16)*x^4-8*x^5)*d/dx P(n-2,x) + (10*n-6+(4*n^2-6*n+22)*x+(10*n-8)*x^2+4*x^3)*P(n-1,x) + ((2*n-2)*x-(2*n-6)*x^2-4*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[(-(24*n + (16*n^2 - 24*n + 48)*x + (4*n^3 - 8*n^2 + 40*n - 48)*x^2 + (12*n^2 - 16*n + 16)*x^3 + 8*n*x^4)*P[n - 2, x] - (8*n*x + (4*n^2 - 12*n + 16)*x^2 - (4*n^2 - 16*n + 24)*x^3 - (12*n - 16)*x^4 - 8*x^5)*D[P[n - 2, x], x] + (10*n - 6 + (4*n^2 - 6*n + 22)*x + (10*n - 8)*x^2 + 4*x^3)*P[n - 1, x] + ((2*n - 2)*x - (2*n - 6)*x^2 - 4*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] = (-(24*n + (16*n^2 - 24*n + 48)*x + (4*n^3 - 8*n^2 + 40*n - 48)*x^2 + (12*n^2 - 16*n + 16)*x^3 + 8*n*x^4)*v[i - 2] - (8*n*x + (4*n^2 - 12*n + 16)*x^2 - (4*n^2 - 16*n + 24)*x^3 - (12*n - 16)*x^4 - 8*x^5)*deriv(v[i - 2], x) + (10*n - 6 + (4*n^2 - 6*n + 22)*x + (10*n - 8)*x^2 + 4*x^3)*v[i - 1] + ((2*n - 2)*x - (2*n - 6)*x^2 - 4*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) = 2*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) = 2*x + 4.?
    
    ↑ Liu–Wang theorem (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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.