A091042 → 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
\(\left(9 - 3\,x - 6\,n + 11\,n x - 12\,n^{2} x + 4\,n^{3} x\right)\,P_{n}(x) = \left(9 + 51\,x + 12\,x^{2} - 6\,n - 25\,n x - 5\,n x^{2} - 12\,n^{2} x - 20\,n^{2} x^{2} + 4\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-12\,x + 6\,x^{2} + 6\,x^{3} + 8\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 6 | 7 | ||||||
| 3 | 1 | 20 | 15 | 36 | |||||
| 4 | 1 | 42 | 105 | 28 | 176 | ||||
| 5 | 1 | 72 | 378 | 336 | 45 | 832 | |||
| 6 | 1 | 110 | 990 | 1848 | 825 | 66 | 3840 | ||
| 7 | 1 | 156 | 2145 | 6864 | 6435 | 1716 | 91 | 17408 | |
| 8 | 1 | 210 | 4095 | 20020 | 32175 | 18018 | 3185 | 120 | 77824 |
Computed in your browser ?
Technical details
- Node id
n-b82f6f873918cdd2· revision 1 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-d402ff27ba37b944
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: 1, 6; 3: 1, 20, 15; 4: 1, 42, 105, 28; 5: 1, 72, 378, 336, 45; 6: 1, 110, 990, 1848, 825, 66; 7: 1, 156, 2145, 6864, 6435, 1716, 91; 8: 1, 210, 4095, 20020, 32175, 18018, 3185, 120; 9: 1, 272, 7140, 49504, 121550, 116688, 43316, 5440, 153; 10: 1, 342, 11628, 108528, 377910, 554268, 352716, 93024, 8721, 190;
%F FormulaThe row polynomials satisfy -(6*n-9-(4*n^3-12*n^2+11*n-3)*x)*P(n,x) = -(6*n-9-(4*n^3-12*n^2-25*n+51)*x-(12*n^3-20*n^2-5*n+12)*x^2)*P(n-1,x) - (12*x-(8*n^2+6)*x^2+(8*n^2-6)*x^3)*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[(-(6*n - 9 - (4*n^3 - 12*n^2 - 25*n + 51)*x - (12*n^3 - 20*n^2 - 5*n + 12)*x^2)*P[n - 1, x] - (12*x - (8*n^2 + 6)*x^2 + (8*n^2 - 6)*x^3)*D[P[n - 1, x], x])/(-6*n + 9 + (4*n^3 - 12*n^2 + 11*n - 3)*x)]];
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] = (-(6*n - 9 - (4*n^3 - 12*n^2 - 25*n + 51)*x - (12*n^3 - 20*n^2 - 5*n + 12)*x^2)*v[i - 1] - (12*x - (8*n^2 + 6)*x^2 + (8*n^2 - 6)*x^3)*deriv(v[i - 1], x))/(-6*n + 9 + (4*n^3 - 12*n^2 + 11*n - 3)*x)); 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-1081279861cd2a9c -
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-1081279861cd2a9c -
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-1081279861cd2a9c -
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-1081279861cd2a9c
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-1081279861cd2a9cLaguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
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premise (sequence):
Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
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theorem_check three_term_recurrence
e-1c0df90858517049root: theorem checkThree-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
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premise (sequence):
Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
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
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023