A156289 → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-3\,x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} - 2\,x^{3}\right)\,P_{n-2}'(x) + \left(-x^{3}\right)\,P_{n-2}''(x) + \left(1 + 6\,x\right)\,P_{n-1}(x) + \left(3\,x + 2\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 6\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 6 | 7 | ||||||
| 3 | 1 | 30 | 45 | 76 | |||||
| 4 | 1 | 126 | 630 | 420 | 1177 | ||||
| 5 | 1 | 510 | 6615 | 12600 | 4725 | 24451 | |||
| 6 | 1 | 2046 | 63360 | 263340 | 259875 | 62370 | 650992 | ||
| 7 | 1 | 8190 | 585585 | 4804800 | 9459450 | 5675670 | 945945 | 21479641 | |
| 8 | 1 | 32766 | 5331690 | 82342260 | 290540250 | 329188860 | 132432300 | 16216200 | 856084327 |
Compute more
?
Technical details
- Node id
n-2f3805888228e572· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-e828929508d533c3
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
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 1, 6; 3: 1, 30, 45; 4: 1, 126, 630, 420; 5: 1, 510, 6615, 12600, 4725; 6: 1, 2046, 63360, 263340, 259875, 62370; 7: 1, 8190, 585585, 4804800, 9459450, 5675670, 945945; 8: 1, 32766, 5331690, 82342260, 290540250, 329188860, 132432300, 16216200; 9: 1, 131070, 48230955, 1367238600, 8189856675, 15685930260, 11481880410, 3308104800, 310134825; 10: 1, 524286, 435061620, 22325973180, 220127851125, 672799597470, 800515415700, 408550942800, 88388425125, 6547290750;
%F FormulaThe row polynomials satisfy P(n,x) = -3*x^2*P(n-2,x) - (2*x^2+2*x^3)*d/dx P(n-2,x) - x^3*d^2/dx^2 P(n-2,x) + (1+6*x)*P(n-1,x) + (3*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 6*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[-3*x^2*P[n - 2, x] - (2*x^2 + 2*x^3)*D[P[n - 2, x], x] - x^3*D[P[n - 2, x], {x, 2}] + (1 + 6*x)*P[n - 1, x] + (3*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 6*x; for(n = 3, N, my(i = n); v[i] = -3*x^2*v[i - 2] - (2*x^2 + 2*x^3)*deriv(v[i - 2], x) - x^3*deriv(deriv(v[i - 2], x), x) + (1 + 6*x)*v[i - 1] + (3*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.