A272866 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + x + x^{2} + 3\,n + 2\,n x\right)\,P_{n}(x) = \left(1 + 6\,x + 11\,x^{2} + 6\,x^{3} + x^{4} + 3\,n + 11\,n x + 9\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	5	1													10
2	7	28	29	10	1											75
3	10	69	165	165	75	15	1									500
4	13	128	490	924	915	492	142	20	1							3125
5	16	205	1085	3070	5050	4983	2995	1090	230	25	1					18750
6	19	300	2031	7710	18060	27192	26795	17418	7440	2040	339	30	1			109375
7	22	413	3409	16275	49791	102396	144914	142747	98315	47208	15582	3423	469	35	1	625000

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-473a01c8b81d3d93 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d530cb8fed751942

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 ?

A053464: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:  4,   5,    1;
  2:  7,  28,   29,    10,      1;
  3: 10,  69,  165,   165,     75,     15,       1;
  4: 13, 128,  490,   924,    915,    492,     142,      20,       1;
  5: 16, 205, 1085,  3070,   5050,   4983,    2995,    1090,     230,      25,       1;
  6: 19, 300, 2031,  7710,  18060,  27192,   26795,   17418,    7440,    2040,     339,      30,      1;
  7: 22, 413, 3409, 16275,  49791, 102396,  144914,  142747,   98315,   47208,   15582,    3423,    469,     35,     1;
  8: 25, 544, 5300, 30520, 115570, 303240,  566888,  766400,  755035,  543560,  285320,  108192,  29050,   5320,   620,   40,   1;
  9: 28, 693, 7785, 52524, 237420, 759906, 1776306, 3087000, 4029228, 3970935, 2960379, 1666584, 703920, 220122, 49770, 7812, 792, 45, 1;

%F Formula

The row polynomials satisfy (3*n-2+(2*n+1)*x+x^2)*P(n,x) = (3*n+1+(11*n+6)*x+(9*n+11)*x^2+(2*n+6)*x^3+x^4)*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[((3*n + 1 + (11*n + 6)*x + (9*n + 11)*x^2 + (2*n + 6)*x^3 + x^4)*P[n - 1, x])/(3*n - 2 + (2*n + 1)*x + x^2)]];
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] = ((3*n + 1 + (11*n + 6)*x + (9*n + 11)*x^2 + (2*n + 6)*x^3 + x^4)*v[i - 1])/(3*n - 2 + (2*n + 1)*x + x^2)); 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-5c9a5bac3deec721
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-5c9a5bac3deec721
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-5c9a5bac3deec721

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-5c9a5bac3deec721

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.