A062145 → 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
\(\left(2 + 6\,x - 2\,x^{2} + n - 7\,n x + n x^{2} - 4\,n^{2} - n^{2} x + n^{2} x^{2} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(2 + 2\,x - 6\,x^{2} + 2\,x^{3} - 3\,n + n x + 3\,n x^{2} + n x^{3} - 2\,n^{2} - 8\,n^{2} x - 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + 2\,n^{3} + 2\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(-x^{2} + 2\,x^{3} - x^{4} - 4\,n x + 4\,n x^{2} + 2\,n x^{3} - 2\,n x^{4} + 2\,n^{2} x + 2\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 5 | 4 | 9 | ||||||
| 3 | 11 | 30 | 10 | 51 | |||||
| 4 | 19 | 108 | 105 | 20 | 252 | ||||
| 5 | 29 | 280 | 546 | 280 | 35 | 1170 | |||
| 6 | 41 | 600 | 1960 | 1960 | 630 | 56 | 5247 | ||
| 7 | 55 | 1134 | 5580 | 9240 | 5670 | 1260 | 84 | 23023 | |
| 8 | 71 | 1960 | 13545 | 33600 | 33810 | 14112 | 2310 | 120 | 99528 |
Computed in your browser ?
Technical details
- Node id
n-efa728d8a00b7c8f· revision 1 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-41d66f02e7ef94b4
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 -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 ?
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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 5, 4; 3: 11, 30, 10; 4: 19, 108, 105, 20; 5: 29, 280, 546, 280, 35; 6: 41, 600, 1960, 1960, 630, 56; 7: 55, 1134, 5580, 9240, 5670, 1260, 84; 8: 71, 1960, 13545, 33600, 33810, 14112, 2310, 120; 9: 89, 3168, 29260, 101640, 152460, 103488, 31416, 3960, 165; 10: 109, 4860, 57816, 267960, 561330, 565488, 277200, 64152, 6435, 220;
%F FormulaThe row polynomials satisfy (n^4-4*n^2+n+2+(2*n^3-n^2-7*n+6)*x+(n^2+n-2)*x^2)*P(n,x) = (n^4+2*n^3-2*n^2-3*n+2+(3*n^4+2*n^3-8*n^2+n+2)*x+(6*n^3-3*n^2+3*n-6)*x^2+(3*n^2+n+2)*x^3)*P(n-1,x) + ((2*n^3+2*n^2-4*n)*x-(2*n^3-2*n^2-4*n+1)*x^2-(4*n^2-2*n-2)*x^3-(2*n+1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 + 2*n^3 - 2*n^2 - 3*n + 2 + (3*n^4 + 2*n^3 - 8*n^2 + n + 2)*x + (6*n^3 - 3*n^2 + 3*n - 6)*x^2 + (3*n^2 + n + 2)*x^3)*P[n - 1, x] + ((2*n^3 + 2*n^2 - 4*n)*x - (2*n^3 - 2*n^2 - 4*n + 1)*x^2 - (4*n^2 - 2*n - 2)*x^3 - (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 4*n^2 + n + 2 + (2*n^3 - n^2 - 7*n + 6)*x + (n^2 + n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^4 + 2*n^3 - 2*n^2 - 3*n + 2 + (3*n^4 + 2*n^3 - 8*n^2 + n + 2)*x + (6*n^3 - 3*n^2 + 3*n - 6)*x^2 + (3*n^2 + n + 2)*x^3)*v[i - 1] + ((2*n^3 + 2*n^2 - 4*n)*x - (2*n^3 - 2*n^2 - 4*n + 1)*x^2 - (4*n^2 - 2*n - 2)*x^3 - (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 4*n^2 + n + 2 + (2*n^3 - n^2 - 7*n + 6)*x + (n^2 + n - 2)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)Generated with polytool c48629374597.
Proved properties
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Real-rooted. Every row P_n, n >= 1, 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-df4a805affbb39de -
Coefficients of one sign. Every nonzero row, n >= 1: 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-df4a805affbb39de -
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-df4a805affbb39de -
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.by the cited steps:
e-df4a805affbb39de
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
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preserver_image derivative_polynomial, shift 0
e-df4a805affbb39deLaguerre’s theorem on differential operators [SP14] polytool ?
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premise (sequence):
Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…
uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
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theorem_check liu_wang
e-62efc507a1cd208aroot: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
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premise (sequence):
Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…
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.