A126351 → 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

\(\left(2\,x + n\right)\,P_{n}(x) = \left(-1 + 3\,x^{2} - 2\,x^{3} - n - 4\,n x^{2} + n x^{3} - 2\,n x^{4} - 2\,n^{2} x - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(x^{2} + 2\,x^{3} + x^{4} + 2\,x^{5} + n x^{2} + n x^{3} + 3\,n x^{4} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 4\,x - x^{2} + 2\,x^{3} + 2\,n + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x^{3} - n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	6	13	4						23
3	10	47	43	8					108
4	15	124	250	129	16				534
5	21	270	980	1129	371	32			2803
6	28	517	2975	6314	4676	1049	64		15623
7	36	903	7574	25984	36246	18423	2955	128	92249

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-11c961c57350750b · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-ac6555d98267de92

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,    2;
  2:  6,   13,     4;
  3: 10,   47,    43,      8;
  4: 15,  124,   250,    129,     16;
  5: 21,  270,   980,   1129,    371,      32;
  6: 28,  517,  2975,   6314,   4676,    1049,     64;
  7: 36,  903,  7574,  25984,  36246,   18423,   2955,    128;
  8: 45, 1472, 16968,  86142, 198051,  193584,  70470,   8353,   256;
  9: 55, 2274, 34500, 243558, 841407, 1381890, 985720, 265013, 23779, 512;

%F Formula

The row polynomials satisfy (n+2*x)*P(n,x) = -(n+1+2*n^2*x+(n^3+4*n-3)*x^2+(3*n^2-n+2)*x^3+2*n*x^4)*P(n-2,x) + ((n+1)*x^2+(n^2+n+2)*x^3+(3*n+1)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n+1+(2*n^2+4)*x+(5*n-1)*x^2+2*x^3)*P(n-1,x) - (n*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + 2*n^2*x + (n^3 + 4*n - 3)*x^2 + (3*n^2 - n + 2)*x^3 + 2*n*x^4)*P[n - 2, x] + ((n + 1)*x^2 + (n^2 + n + 2)*x^3 + (3*n + 1)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n + 1 + (2*n^2 + 4)*x + (5*n - 1)*x^2 + 2*x^3)*P[n - 1, x] - (n*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + 2*n^2*x + (n^3 + 4*n - 3)*x^2 + (3*n^2 - n + 2)*x^3 + 2*n*x^4)*v[i - 2] + ((n + 1)*x^2 + (n^2 + n + 2)*x^3 + (3*n + 1)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n + 1 + (2*n^2 + 4)*x + (5*n - 1)*x^2 + 2*x^3)*v[i - 1] - (n*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 2*x)); 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-60a7391d38865b67
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-60a7391d38865b67
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-60a7391d38865b67
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-60a7391d38865b67

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-60a7391d38865b67

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-0c08584d383e1215 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.