A084950 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-4}(x) + \left(2\,x - 2\,n x\right)\,P_{n-3}(x) + \left(2\,x + n - n^{2}\right)\,P_{n-2}(x) + \left(2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
\(P_{2}(x) = 10 + 4\,x\)
\(P_{3}(x) = 42 + 20\,x + x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 1 | 4 | ||||||
| 2 | 10 | 4 | 14 | ||||||
| 3 | 42 | 20 | 1 | 63 | |||||
| 4 | 216 | 114 | 9 | 339 | |||||
| 5 | 1320 | 744 | 75 | 1 | 2140 | ||||
| 6 | 9360 | 5520 | 648 | 16 | 15544 | ||||
| 7 | 75600 | 46080 | 6000 | 204 | 1 | 127885 | |||
| 8 | 685440 | 428400 | 60120 | 2500 | 25 | 1176485 | |||
| 9 | 6894720 | 4394880 | 652680 | 31200 | 455 | 1 | 11973936 | ||
| 10 | 76204800 | 49351680 | 7660800 | 405720 | 7530 | 36 | 133630566 | ||
| 11 | 918086400 | 602380800 | 96888960 | 5550720 | 122010 | 888 | 1 | 1623029779 | |
| 12 | 11975040000 | 7943443200 | 1315440000 | 80196480 | 1999200 | 19110 | 49 | 21316138039 | |
| 13 | 168129561600 | 112565376000 | 19100188800 | 1224720000 | 33657120 | 390432 | 1575 | 1 | 301053895528 |
| 14 | 2528170444800 | 1706203699200 | 295543987200 | 19758816000 | 586958400 | 7874496 | 42784 | 64 | 4550271822944 |
Compute more
?
Technical details
- Node id
n-78b9be25caf5ba66· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-f046eb9fd944bb9d
Observed on the stored rows ?
- Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (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: 1; 1: 3, 1; 2: 10, 4; 3: 42, 20, 1; 4: 216, 114, 9; 5: 1320, 744, 75, 1; 6: 9360, 5520, 648, 16; 7: 75600, 46080, 6000, 204, 1; 8: 685440, 428400, 60120, 2500, 25; 9: 6894720, 4394880, 652680, 31200, 455, 1;
%F FormulaThe row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*n-2)*x*P(n-3,x) - (n^2-n-2*x)*P(n-2,x) + 2*n*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + x, P(2,x) = 10 + 4*x, P(3,x) = 42 + 20*x + x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 3 + x;
P[2, x_] := 10 + 4*x;
P[3, x_] := 42 + 20*x + x^2;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*n - 2)*x*P[n - 3, x] - (n^2 - n - 2*x)*P[n - 2, x] + 2*n*P[n - 1, 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] = 3 + x; v[3] = 10 + 4*x; v[4] = 42 + 20*x + x^2; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - (2*n - 2)*x*v[i - 3] - (n^2 - n - 2*x)*v[i - 2] + 2*n*v[i - 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
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.