A121574 → D+1

nonnegative coefficients

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

Recurrence

\(\left(28\,n x - 15\,n x^{2} + 3\,n x^{3} + n x^{4} + 6\,n^{2} - 11\,n^{2} x + 5\,n^{2} x^{2} + n^{2} x^{3} - 6\,n^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(-28\,x^{2} + 20\,x^{3} + 10\,x^{4} + x^{5} + 12\,n + 70\,n x + 41\,n x^{2} - 14\,n x^{3} - 3\,n x^{4} + 2\,n^{2} x - 23\,n^{2} x^{2} - 4\,n^{2} x^{3} - 12\,n^{3} - 12\,n^{3} x\right)\,P_{n-1}(x) + \left(28\,x^{3} + 60\,x^{4} + 15\,x^{5} + x^{6} + 12\,n x + 52\,n x^{2} + 63\,n x^{3} + 15\,n x^{4} + n x^{5} + 12\,n^{2} x + 24\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	1							4
2	9	7	1						17
3	24	32	11	1					68
4	60	118	70	15	1				264
5	144	380	335	123	19	1			1002
6	336	1120	1339	729	191	23	1		3739
7	768	3104	4736	3504	1354	274	27	1	13768

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-2ffb0ff2c4cf68d9 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-422d7b1e2dc04ac9

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    1;
  1:    3,     1;
  2:    9,     7,     1;
  3:   24,    32,    11,     1;
  4:   60,   118,    70,    15,     1;
  5:  144,   380,   335,   123,    19,     1;
  6:  336,  1120,  1339,   729,   191,    23,    1;
  7:  768,  3104,  4736,  3504,  1354,   274,   27,   1;
  8: 1728,  8224, 15352, 14556,  7631,  2264,  372,  31,  1;
  9: 3840, 21056, 46640, 54360, 36547, 14665, 3513, 485, 35, 1;

%F Formula

The row polynomials satisfy -(6*n^3-6*n^2-(3*n^3-11*n^2+28*n)*x-(5*n^2-15*n)*x^2-(n^2+3*n)*x^3-n*x^4)*P(n,x) = -(12*n^3-12*n+(12*n^3-2*n^2-70*n)*x+(23*n^2-41*n+28)*x^2+(4*n^2+14*n-20)*x^3+(3*n-10)*x^4-x^5)*P(n-1,x) + ((12*n^2+12*n)*x+(24*n^2+52*n)*x^2+(3*n^2+63*n+28)*x^3+(15*n+60)*x^4+(n+15)*x^5+x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n^3 - 12*n + (12*n^3 - 2*n^2 - 70*n)*x + (23*n^2 - 41*n + 28)*x^2 + (4*n^2 + 14*n - 20)*x^3 + (3*n - 10)*x^4 - x^5)*P[n - 1, x] + ((12*n^2 + 12*n)*x + (24*n^2 + 52*n)*x^2 + (3*n^2 + 63*n + 28)*x^3 + (15*n + 60)*x^4 + (n + 15)*x^5 + x^6)*D[P[n - 1, x], x])/(-6*n^3 + 6*n^2 + (3*n^3 - 11*n^2 + 28*n)*x + (5*n^2 - 15*n)*x^2 + (n^2 + 3*n)*x^3 + n*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(12*n^3 - 12*n + (12*n^3 - 2*n^2 - 70*n)*x + (23*n^2 - 41*n + 28)*x^2 + (4*n^2 + 14*n - 20)*x^3 + (3*n - 10)*x^4 - x^5)*v[i - 1] + ((12*n^2 + 12*n)*x + (24*n^2 + 52*n)*x^2 + (3*n^2 + 63*n + 28)*x^3 + (15*n + 60)*x^4 + (n + 15)*x^5 + x^6)*deriv(v[i - 1], x))/(-6*n^3 + 6*n^2 + (3*n^3 - 11*n^2 + 28*n)*x + (5*n^2 - 15*n)*x^2 + (n^2 + 3*n)*x^3 + n*x^4)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-b8ffc6ab03d37d48
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-b8ffc6ab03d37d48
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-b8ffc6ab03d37d48
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-b8ffc6ab03d37d48

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image derivative_polynomial, shift 0 e-b8ffc6ab03d37d48

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-81aa6611c032ec70 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

No candidate equalities or related sequences.

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.