A272866 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	3	2													5
2	6	22	18	4											50
3	9	60	135	120	45	6									375
4	12	116	432	780	720	348	84	8							2500
5	15	190	990	2740	4365	4110	2310	760	135	10					15625
6	18	282	1890	7080	16290	23934	22806	14160	5670	1410	198	12			93750
7	21	392	3213	15204	45990	93198	129381	124264	82782	38010	11781	2352	273	14	546875

Show all 15 stored rows

Generated b-files: A036291 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-b60ac0f95ba69845 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-0e4ed98624db992d

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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 ?

A036291: row sums, 15 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;
  2:  6,  22,   18,     4;
  3:  9,  60,  135,   120,     45,      6;
  4: 12, 116,  432,   780,    720,    348,      84,       8;
  5: 15, 190,  990,  2740,   4365,   4110,    2310,     760,     135,      10;
  6: 18, 282, 1890,  7080,  16290,  23934,   22806,   14160,    5670,    1410,     198,      12;
  7: 21, 392, 3213, 15204,  45990,  93198,  129381,  124264,   82782,   38010,   11781,    2352,    273,     14;
  8: 24, 520, 5040, 28840, 108360, 281568,  519960,  692120,  668520,  469280,  238392,   86520,  21840,   3640,   360,   16;
  9: 27, 666, 7452, 50040, 224910, 714924, 1657152, 2850264, 3672945, 3562830, 2604096, 1429848, 584766, 175140, 37260, 5328, 459, 18;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 3*n*x + n*x^2)*P[n - 1, x])/(n - 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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + 3*n*x + n*x^2)*v[i - 1])/(n - 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-5fb206b09638d5b7
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-5fb206b09638d5b7
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-5fb206b09638d5b7

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-5fb206b09638d5b7

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image reversal, shift 0 e-0e058c428dfdfeb6
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-d82c9b48e4b31a6b root: theorem check
    
    Parent-row provers of polytool polytool ?
  - ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

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