A201637 → reverse (shift +1) → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(1 - 2\,x + 2\,n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||
| 1 | 1 | 1 | ||||||
| 2 | 1 | 2 | 3 | |||||
| 3 | 1 | 8 | 6 | 15 | ||||
| 4 | 1 | 22 | 58 | 24 | 105 | |||
| 5 | 1 | 52 | 328 | 444 | 120 | 945 | ||
| 6 | 1 | 114 | 1452 | 4400 | 3708 | 720 | 10395 | |
| 7 | 1 | 240 | 5610 | 32120 | 58140 | 33984 | 5040 | 135135 |
Compute more
?
Technical details
- Node id
n-db6c83a2e7fef454· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-36df5d4b0f5901bf
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- 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; 2: 1, 2; 3: 1, 8, 6; 4: 1, 22, 58, 24; 5: 1, 52, 328, 444, 120; 6: 1, 114, 1452, 4400, 3708, 720; 7: 1, 240, 5610, 32120, 58140, 33984, 5040; 8: 1, 494, 19950, 195800, 644020, 785304, 341136, 40320; 9: 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880;
%F FormulaThe row polynomials satisfy P(n,x) = (1+(2*n-2)*x)*P(n-1,x) + (x-x^2)*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[(1 + (2*n - 2)*x)*P[n - 1, x] + (x - 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; for(n = 1, N, my(i = n + 1); v[i] = (1 + (2*n - 2)*x)*v[i - 1] + (x - 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)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
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
Related ?
Same fingerprint ?
- A163936: Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).
- A288874: Row reversed version of triangle A201637 (second-order Eulerian triangle).
- A340556: E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <=…
- A008517: Second-order Eulerian triangle T(n,k), 1 <= k <= n.
- A201637: Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
- A112007: Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.
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