A249248 → θ+1
nonnegative coefficients
Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means
Recurrence
\(\left(1 + n\right)\,P_{n}(x) = \left(2\,n x^{2} - n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-4}(x) + \left(-4\,n x - 3\,n x^{2} - 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-2 - 2\,x - x^{2} - n + 2\,n x - n x^{2} + 2\,n^{2} x\right)\,P_{n-2}(x) + \left(3 + 2\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 6\,x\)
\(P_{2}(x) = 1 + 14\,x + 9\,x^{2}\)
\(P_{3}(x) = 1 + 24\,x + 66\,x^{2} + 12\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 6 | 7 | ||||||
| 2 | 1 | 14 | 9 | 24 | |||||
| 3 | 1 | 24 | 66 | 12 | 103 | ||||
| 4 | 1 | 36 | 207 | 160 | 15 | 419 | |||
| 5 | 1 | 50 | 477 | 964 | 305 | 18 | 1815 | ||
| 6 | 1 | 66 | 930 | 3532 | 2910 | 510 | 21 | 7970 | |
| 7 | 1 | 84 | 1629 | 9860 | 16965 | 6930 | 784 | 24 | 36277 |
Computed in your browser ?
Technical details
- Node id
n-5e13b8dd79cb0234· revision 1 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-90a572ac0a972444
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 ExampleTriangle begins: 0: 1; 1: 1, 6; 2: 1, 14, 9; 3: 1, 24, 66, 12; 4: 1, 36, 207, 160, 15; 5: 1, 50, 477, 964, 305, 18; 6: 1, 66, 930, 3532, 2910, 510, 21; 7: 1, 84, 1629, 9860, 16965, 6930, 784, 24; 8: 1, 104, 2646, 23192, 69025, 58716, 14224, 1136, 27; 9: 1, 126, 4062, 48440, 221265, 345126, 163576, 26352, 1575, 30;
%F FormulaThe row polynomials satisfy (n+1)*P(n,x) = -(n^3+n^2-2*n)*x^2*P(n-4,x) - ((2*n^2+4*n)*x+(2*n^2+3*n)*x^2)*P(n-3,x) - (n+2-(2*n^2+2*n-2)*x+(n+1)*x^2)*P(n-2,x) + (2*n+3+(2*n+2)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 6*x, P(2,x) = 1 + 14*x + 9*x^2, P(3,x) = 1 + 24*x + 66*x^2 + 12*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 6*x;
P[2, x_] := 1 + 14*x + 9*x^2;
P[3, x_] := 1 + 24*x + 66*x^2 + 12*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + n^2 - 2*n)*x^2*P[n - 4, x] - ((2*n^2 + 4*n)*x + (2*n^2 + 3*n)*x^2)*P[n - 3, x] - (n + 2 - (2*n^2 + 2*n - 2)*x + (n + 1)*x^2)*P[n - 2, x] + (2*n + 3 + (2*n + 2)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 6*x; v[3] = 1 + 14*x + 9*x^2; v[4] = 1 + 24*x + 66*x^2 + 12*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 + n^2 - 2*n)*x^2*v[i - 4] - ((2*n^2 + 4*n)*x + (2*n^2 + 3*n)*x^2)*v[i - 3] - (n + 2 - (2*n^2 + 2*n - 2)*x + (n + 1)*x^2)*v[i - 2] + (2*n + 3 + (2*n + 2)*x)*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
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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-984e895a2b17c62b -
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-984e895a2b17c62b -
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-984e895a2b17c62b -
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-984e895a2b17c62b
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 euler_polynomial, shift 0
e-984e895a2b17c62bLaguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
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premise (sequence):
Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
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theorem_check liu_wang_derivative_lag
e-2dcae0cd92b4b84aroot: theorem checkLiu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
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premise (sequence):
Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
Related sequences ?
No candidate equalities or related sequences.
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [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.