A060187 → γ → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(n - 24\,n x - 48\,n x^{2} + 8\,n^{2} x + 64\,n^{2} x^{2} - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,n x - 32\,n x^{3} - 8\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-1 + 16\,x - 24\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 8\,x^{2} + 2\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 1 | 1 | 1 | |||
| 2 | 1 | 1 | |||
| 3 | 1 | 8 | 9 | ||
| 4 | 1 | 40 | 41 | ||
| 5 | 1 | 144 | 240 | 385 | |
| 6 | 1 | 464 | 2928 | 3393 | |
| 7 | 1 | 1432 | 22992 | 15616 | 40041 |
| 8 | 1 | 4344 | 149328 | 354560 | 508233 |
Compute more
?
Technical details
- Node id
n-65a31461f7438a24· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-ade430673929d22a
Observed on the stored rows ?
- Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 1; 3: 1, 8; 4: 1, 40; 5: 1, 144, 240; 6: 1, 464, 2928; 7: 1, 1432, 22992, 15616; 8: 1, 4344, 149328, 354560; 9: 1, 13088, 876960, 4871168, 1772800; 10: 1, 39328, 4855968, 52806656, 64666880;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (n+(8*n^2-24*n)*x-(16*n^3-64*n^2+48*n)*x^2)*P(n-2,x) + (2*n*x-8*n^2*x^2+(32*n^2-32*n)*x^3)*d/dx P(n-2,x) - (1-(8*n^2-24*n+16)*x)*P(n-1,x) + ((2*n-2)*x-(8*n-8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + (8*n^2 - 24*n)*x - (16*n^3 - 64*n^2 + 48*n)*x^2)*P[n - 2, x] + (2*n*x - 8*n^2*x^2 + (32*n^2 - 32*n)*x^3)*D[P[n - 2, x], x] - (1 - (8*n^2 - 24*n + 16)*x)*P[n - 1, x] + ((2*n - 2)*x - (8*n - 8)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n + (8*n^2 - 24*n)*x - (16*n^3 - 64*n^2 + 48*n)*x^2)*v[i - 2] + (2*n*x - 8*n^2*x^2 + (32*n^2 - 32*n)*x^3)*deriv(v[i - 2], x) - (1 - (8*n^2 - 24*n + 16)*x)*v[i - 1] + ((2*n - 2)*x - (8*n - 8)*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, 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]
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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
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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
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from gamma_polynomial of Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) =…?
Related ?
Same fingerprint ?
References
- [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.