A049029 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(1 - 5\,x + 5\,n x\right)\,P_{n-1}(x) + \left(-x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 5 | 6 | ||||||
| 3 | 1 | 15 | 45 | 61 | |||||
| 4 | 1 | 30 | 255 | 585 | 871 | ||||
| 5 | 1 | 50 | 825 | 5175 | 9945 | 15996 | |||
| 6 | 1 | 75 | 2025 | 24150 | 123795 | 208845 | 358891 | ||
| 7 | 1 | 105 | 4200 | 80850 | 775845 | 3427515 | 5221125 | 9509641 | |
| 8 | 1 | 140 | 7770 | 219450 | 3363045 | 27478710 | 108046575 | 151412625 | 290528316 |
Compute more
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Technical details
- Node id
n-ef665de83611fdca· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-e050e5a67817afb8
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, 5; 3: 1, 15, 45; 4: 1, 30, 255, 585; 5: 1, 50, 825, 5175, 9945; 6: 1, 75, 2025, 24150, 123795, 208845; 7: 1, 105, 4200, 80850, 775845, 3427515, 5221125; 8: 1, 140, 7770, 219450, 3363045, 27478710, 108046575, 151412625; 9: 1, 180, 13230, 514710, 11482695, 148548330, 1069801425, 3824996175, 4996616625; 10: 1, 225, 21150, 1083600, 33100515, 619338825, 7011734625, 45547251750, 150346471275, 184874815125;
%F FormulaThe row polynomials satisfy P(n,x) = (1+(5*n-5)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (5*n - 5)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (5*n - 5)*x)*v[i - 1] - 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, 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
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
Related ?
Same fingerprint ?
- A049029: Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.
- A225471 → basis A008277 → D+1
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
- [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.