A110552 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(2 + 2\,n x - n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 12\,x - 4\,x^{3} - 2\,n x + 11\,n x^{2} + 2\,n x^{3} - 3\,n^{2} x^{2} + 3\,n^{2} x^{3} - n^{3} x^{3}\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{3} - 2\,x^{4} + 3\,n x^{2} + 3\,n x^{3} + n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 4 | 5 | ||||||
| 2 | 1 | 10 | 12 | 23 | |||||
| 3 | 1 | 20 | 51 | 32 | 104 | ||||
| 4 | 1 | 38 | 153 | 196 | 80 | 468 | |||
| 5 | 1 | 72 | 402 | 784 | 645 | 192 | 2096 | ||
| 6 | 1 | 138 | 990 | 2600 | 3225 | 1926 | 448 | 9328 | |
| 7 | 1 | 268 | 2349 | 7760 | 12875 | 11556 | 5383 | 1024 | 41216 |
Compute more
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Technical details
- Node id
n-35912eee1e284d52· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-0733988374790ceb
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
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 4; 2: 1, 10, 12; 3: 1, 20, 51, 32; 4: 1, 38, 153, 196, 80; 5: 1, 72, 402, 784, 645, 192; 6: 1, 138, 990, 2600, 3225, 1926, 448; 7: 1, 268, 2349, 7760, 12875, 11556, 5383, 1024; 8: 1, 526, 5439, 21644, 44975, 53886, 37681, 14344, 2304; 9: 1, 1040, 12372, 57568, 143710, 215376, 200900, 114752, 36873, 5120;
%F FormulaThe row polynomials satisfy (2+2*n*x+(n^2-n)*x^2)*P(n,x) = (2-(2*n-12)*x-(3*n^2-11*n)*x^2-(n^3-3*n^2-2*n+4)*x^3)*P(n-1,x) + (2*x+3*n*x^2+(n^2+3*n-4)*x^3+(n^2-2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2 - (2*n - 12)*x - (3*n^2 - 11*n)*x^2 - (n^3 - 3*n^2 - 2*n + 4)*x^3)*P[n - 1, x] + (2*x + 3*n*x^2 + (n^2 + 3*n - 4)*x^3 + (n^2 - 2)*x^4)*D[P[n - 1, x], x])/(2 + 2*n*x + (n^2 - n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2 - (2*n - 12)*x - (3*n^2 - 11*n)*x^2 - (n^3 - 3*n^2 - 2*n + 4)*x^3)*v[i - 1] + (2*x + 3*n*x^2 + (n^2 + 3*n - 4)*x^3 + (n^2 - 2)*x^4)*deriv(v[i - 1], x))/(2 + 2*n*x + (n^2 - n)*x^2)); 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)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
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.