A102662 → (θ+1)²
nonnegative coefficients
Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means
Recurrence
\(P_{n}(x) = \left(9\,x^{2}\right)\,P_{n-4}(x) + \left(4\,x^{3}\right)\,P_{n-4}'(x) + \left(-4\,x + 18\,x^{2}\right)\,P_{n-3}(x) + \left(4\,x^{2} + 8\,x^{3}\right)\,P_{n-3}'(x) + \left(-1 + 9\,x^{2}\right)\,P_{n-2}(x) + \left(4\,x^{3}\right)\,P_{n-2}'(x) + \left(2 + 4\,x\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 12\,x\)
\(P_{2}(x) = 1 + 20\,x + 27\,x^{2}\)
\(P_{3}(x) = 1 + 28\,x + 99\,x^{2} + 48\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 12 | 13 | ||||||
| 2 | 1 | 20 | 27 | 48 | |||||
| 3 | 1 | 28 | 99 | 48 | 176 | ||||
| 4 | 1 | 36 | 207 | 272 | 75 | 591 | |||
| 5 | 1 | 44 | 351 | 816 | 575 | 108 | 1895 | ||
| 6 | 1 | 52 | 531 | 1808 | 2275 | 1044 | 147 | 5858 | |
| 7 | 1 | 60 | 747 | 3376 | 6375 | 5148 | 1715 | 192 | 17614 |
Computed in your browser ?
Technical details
- Node id
n-1ab9c1cd3df94b60· revision 1 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-aea6b96d325f72ce
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, 12; 2: 1, 20, 27; 3: 1, 28, 99, 48; 4: 1, 36, 207, 272, 75; 5: 1, 44, 351, 816, 575, 108; 6: 1, 52, 531, 1808, 2275, 1044, 147; 7: 1, 60, 747, 3376, 6375, 5148, 1715, 192; 8: 1, 68, 999, 5648, 14475, 17604, 10143, 2624, 243; 9: 1, 76, 1287, 8752, 28575, 47628, 41111, 18112, 3807, 300;
%F FormulaThe row polynomials satisfy P(n,x) = 9*x^2*P(n-4,x) + 4*x^3*d/dx P(n-4,x) - (4*x-18*x^2)*P(n-3,x) + (4*x^2+8*x^3)*d/dx P(n-3,x) - (1-9*x^2)*P(n-2,x) + 4*x^3*d/dx P(n-2,x) + (2+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 12*x, P(2,x) = 1 + 20*x + 27*x^2, P(3,x) = 1 + 28*x + 99*x^2 + 48*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 12*x;
P[2, x_] := 1 + 20*x + 27*x^2;
P[3, x_] := 1 + 28*x + 99*x^2 + 48*x^3;
P[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 4, x] + 4*x^3*D[P[n - 4, x], x] - (4*x - 18*x^2)*P[n - 3, x] + (4*x^2 + 8*x^3)*D[P[n - 3, x], x] - (1 - 9*x^2)*P[n - 2, x] + 4*x^3*D[P[n - 2, x], x] + (2 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];
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 + 12*x; v[3] = 1 + 20*x + 27*x^2; v[4] = 1 + 28*x + 99*x^2 + 48*x^3; for(n = 4, N, my(i = n + 1); v[i] = 9*x^2*v[i - 4] + 4*x^3*deriv(v[i - 4], x) - (4*x - 18*x^2)*v[i - 3] + (4*x^2 + 8*x^3)*deriv(v[i - 3], x) - (1 - 9*x^2)*v[i - 2] + 4*x^3*deriv(v[i - 2], x) + (2 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], 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
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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-eaf3df36dd50a693 -
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-eaf3df36dd50a693 -
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-eaf3df36dd50a693 -
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-eaf3df36dd50a693
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-eaf3df36dd50a693Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
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premise (sequence):
Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).
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-5b6052467d550b8droot: theorem checkLiu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
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premise (sequence):
Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).
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.