A228955 → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(-2\,x + n - n x + n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 0 | ||||
| 1 | 1 | 1 | |||
| 2 | 6 | 6 | |||
| 3 | 36 | 12 | 48 | ||
| 4 | 240 | 240 | 480 | ||
| 5 | 1800 | 3600 | 360 | 5760 | |
| 6 | 15120 | 50400 | 15120 | 80640 | |
| 7 | 141120 | 705600 | 423360 | 20160 | 1290240 |
Compute more
?
Technical details
- Node id
n-131f59264e77673e· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-937db0732045a112
Observed on the stored rows ?
- Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half 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 = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 0; 1: 1; 2: 6; 3: 36, 12; 4: 240, 240; 5: 1800, 3600, 360; 6: 15120, 50400, 15120; 7: 141120, 705600, 423360, 20160; 8: 1451520, 10160640, 10160640, 1451520; 9: 16329600, 152409600, 228614400, 65318400, 1814400;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (n^2+n+(n^2-n-2)*x)*P(n-1,x) + ((2*n+2)*x-(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + (n^2 - n - 2)*x)*P[n - 1, x] + ((2*n + 2)*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + n + (n^2 - n - 2)*x)*v[i - 1] + ((2*n + 2)*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(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)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 differential operators
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Proof chains ? JSON
Related ?
Same fingerprint ?
- A003506 → Veronese 2 (j=1)
- A178821 → Veronese 2 → D
- A119900: Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0…
- A178820 → Veronese 2 (j=1)
- A135838 → Veronese 2 (j=1)
- A003506 → Veronese 2 → D
- A178821 → Veronese 2 (j=1)
- A178822 → Veronese 2 → D
- A178820 → Veronese 2 → D
- A178822 → Veronese 2 (j=1)
- A034839 → D
- basis-map coefficients of A113214, shifted → Veronese 2 → D
- A159854 → Veronese 2 (j=1)
- A038208 → Veronese 2 → D
- A038208 → Veronese 2 (j=1)
- A028326 → Veronese 2 (j=1)
- A344565 → Veronese 2 (j=1)
- basis-map coefficients of A113214, shifted → Veronese 2 (j=1)
- A007318 → Veronese 2 (j=1)
- A109447: Binomial coefficients C(n,k) with n-k odd, read by rows.
- A034867: Triangle of odd-numbered terms in rows of Pascal's triangle.
References
- [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.