A062196 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	3							3
2	8	12						20
3	15	60	30					105
4	24	180	240	60				504
5	35	420	1050	700	105			2310
6	48	840	3360	4200	1680	168		10296
7	63	1512	8820	17640	13230	3528	252	45045

Show all 15 stored rows

Generated b-files: A000917 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-82cc9c378f27e13e · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-4e2cf4a8bda0ea40

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

OEIS matches ?

A000917: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:  0;
  1:  3;
  2:  8,   12;
  3: 15,   60,    30;
  4: 24,  180,   240,     60;
  5: 35,  420,  1050,    700,    105;
  6: 48,  840,  3360,   4200,   1680,    168;
  7: 63, 1512,  8820,  17640,  13230,   3528,   252;
  8: 80, 2520, 20160,  58800,  70560,  35280,  6720,   360;
  9: 99, 3960, 41580, 166320, 291060, 232848, 83160, 11880, 495;

%F Formula

The row polynomials satisfy (n^2-1)*P(n,x) = (n^2+2*n+3*n^2*x)*P(n-1,x) + ((2*n+1)*x-(2*n+1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n + 3*n^2*x)*P[n - 1, x] + ((2*n + 1)*x - (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 - 1)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 2*n + 3*n^2*x)*v[i - 1] + ((2*n + 1)*x - (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 - 1)); 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-7729d1773ce70fbd
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-7729d1773ce70fbd
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-7729d1773ce70fbd
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-7729d1773ce70fbd

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-7729d1773ce70fbd

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-32e7e43d5a9debda root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

Same fingerprint (7)

Equal up to content, sign, reversal and a small index shift.

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.