A062264 → 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

\(\left(-8\,n + 4\,n x^{2} + 6\,n^{2} + 8\,n^{2} x + n^{2} x^{2} + 6\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(12\,x^{2} + 12\,x^{3} + 4\,n + 24\,n x + 38\,n x^{2} + 10\,n x^{3} + 17\,n^{2} + 38\,n^{2} x + 24\,n^{2} x^{2} + 3\,n^{2} x^{3} + 8\,n^{3} + 20\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(8\,x^{2} - 4\,x^{3} - 4\,x^{4} + 12\,n x - 10\,n x^{3} - 2\,n x^{4} + 11\,n^{2} x - 7\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	6	5							11
2	13	42	15						70
3	22	147	168	35					372
4	33	368	840	504	70				1815
5	46	765	2880	3360	1260	126			8437
6	61	1410	7875	15000	10710	2772	210		38038
7	78	2387	18480	51975	60060	29106	5544	330	167960

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-7a8b80d9cd64a57e · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-40c37eac13214aad

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:   6,    5;
  2:  13,   42,    15;
  3:  22,  147,   168,     35;
  4:  33,  368,   840,    504,     70;
  5:  46,  765,  2880,   3360,   1260,     126;
  6:  61, 1410,  7875,  15000,  10710,    2772,    210;
  7:  78, 2387, 18480,  51975,  60060,   29106,   5544,    330;
  8:  97, 3792, 38808, 150920, 256410,  199584,  70224,  10296,   495;
  9: 118, 5733, 74880, 384384, 900900, 1027026, 576576, 154440, 18018, 715;

%F Formula

The row polynomials satisfy (n^4+6*n^3+6*n^2-8*n+(2*n^3+8*n^2)*x+(n^2+4*n)*x^2)*P(n,x) = (n^4+8*n^3+17*n^2+4*n+(3*n^4+20*n^3+38*n^2+24*n)*x+(6*n^3+24*n^2+38*n+12)*x^2+(3*n^2+10*n+12)*x^3)*P(n-1,x) + ((2*n^3+11*n^2+12*n)*x-(2*n^3+7*n^2-8)*x^2-(4*n^2+10*n+4)*x^3-(2*n+4)*x^4)*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[((n^4 + 8*n^3 + 17*n^2 + 4*n + (3*n^4 + 20*n^3 + 38*n^2 + 24*n)*x + (6*n^3 + 24*n^2 + 38*n + 12)*x^2 + (3*n^2 + 10*n + 12)*x^3)*P[n - 1, x] + ((2*n^3 + 11*n^2 + 12*n)*x - (2*n^3 + 7*n^2 - 8)*x^2 - (4*n^2 + 10*n + 4)*x^3 - (2*n + 4)*x^4)*D[P[n - 1, x], x])/(n^4 + 6*n^3 + 6*n^2 - 8*n + (2*n^3 + 8*n^2)*x + (n^2 + 4*n)*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] = ((n^4 + 8*n^3 + 17*n^2 + 4*n + (3*n^4 + 20*n^3 + 38*n^2 + 24*n)*x + (6*n^3 + 24*n^2 + 38*n + 12)*x^2 + (3*n^2 + 10*n + 12)*x^3)*v[i - 1] + ((2*n^3 + 11*n^2 + 12*n)*x - (2*n^3 + 7*n^2 - 8)*x^2 - (4*n^2 + 10*n + 4)*x^3 - (2*n + 4)*x^4)*deriv(v[i - 1], x))/(n^4 + 6*n^3 + 6*n^2 - 8*n + (2*n^3 + 8*n^2)*x + (n^2 + 4*n)*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-e09e0bff614306ec
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-e09e0bff614306ec
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-e09e0bff614306ec
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-e09e0bff614306ec

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-e09e0bff614306ec

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Coefficient triangle of certain polynomials N(4; m,x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-c22f7d753efa18e6 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

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.