A008517 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-3 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(2 - 11\,x + 23\,x^{2} - 20\,x^{3} + 4\,x^{4} - 4\,n + 18\,n x - 38\,n x^{2} + 30\,n x^{3} - 4\,n x^{4} - 8\,n^{2} x + 28\,n^{2} x^{2} - 12\,n^{2} x^{3} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(x - 6\,x^{2} + 12\,x^{3} - 9\,x^{4} + 2\,x^{5} - 2\,n x + 10\,n x^{2} - 14\,n x^{3} + 6\,n x^{4} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-7 + 15\,x - 11\,x^{2} + 2\,x^{3} + 6\,n - 18\,n x + 10\,n x^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x + 5\,x^{2} - 2\,x^{3} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 2 | 5 | ||||||
| 2 | 9 | 20 | 6 | 35 | |||||
| 3 | 23 | 138 | 130 | 24 | 315 | ||||
| 4 | 53 | 708 | 1660 | 924 | 120 | 3465 | |||
| 5 | 115 | 3018 | 14652 | 19232 | 7308 | 720 | 45045 | ||
| 6 | 241 | 11460 | 101970 | 264680 | 228060 | 64224 | 5040 | 675675 | |
| 7 | 495 | 40394 | 607350 | 2771880 | 4570540 | 2832120 | 623376 | 40320 | 11486475 |
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Technical details
- Node id
n-45531665af5787a9· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-7af5d5c3f437902e
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 0)
- 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, 2; 2: 9, 20, 6; 3: 23, 138, 130, 24; 4: 53, 708, 1660, 924, 120; 5: 115, 3018, 14652, 19232, 7308, 720; 6: 241, 11460, 101970, 264680, 228060, 64224, 5040; 7: 495, 40394, 607350, 2771880, 4570540, 2832120, 623376, 40320; 8: 1005, 135524, 3254760, 24124500, 67965820, 78597840, 37163736, 6636960, 362880; 9: 2027, 439722, 16197328, 184386160, 821551920, 1588786808, 1374213288, 516899232, 76998240, 3628800;
%F FormulaThe row polynomials satisfy (2*n-3+2*x)*P(n,x) = -(4*n-2+(8*n^2-18*n+11)*x+(8*n^3-28*n^2+38*n-23)*x^2+(12*n^2-30*n+20)*x^3+(4*n-4)*x^4)*P(n-2,x) - ((2*n-1)*x+(4*n^2-10*n+6)*x^2-(4*n^2-14*n+12)*x^3-(6*n-9)*x^4-2*x^5)*d/dx P(n-2,x) + (6*n-7+(8*n^2-18*n+15)*x+(10*n-11)*x^2+2*x^3)*P(n-1,x) + ((2*n-3)*x-(2*n-5)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 2 + (8*n^2 - 18*n + 11)*x + (8*n^3 - 28*n^2 + 38*n - 23)*x^2 + (12*n^2 - 30*n + 20)*x^3 + (4*n - 4)*x^4)*P[n - 2, x] - ((2*n - 1)*x + (4*n^2 - 10*n + 6)*x^2 - (4*n^2 - 14*n + 12)*x^3 - (6*n - 9)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (6*n - 7 + (8*n^2 - 18*n + 15)*x + (10*n - 11)*x^2 + 2*x^3)*P[n - 1, x] + ((2*n - 3)*x - (2*n - 5)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(2*n - 3 + 2*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 2 + (8*n^2 - 18*n + 11)*x + (8*n^3 - 28*n^2 + 38*n - 23)*x^2 + (12*n^2 - 30*n + 20)*x^3 + (4*n - 4)*x^4)*v[i - 2] - ((2*n - 1)*x + (4*n^2 - 10*n + 6)*x^2 - (4*n^2 - 14*n + 12)*x^3 - (6*n - 9)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (6*n - 7 + (8*n^2 - 18*n + 15)*x + (10*n - 11)*x^2 + 2*x^3)*v[i - 1] + ((2*n - 3)*x - (2*n - 5)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(2*n - 3 + 2*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]
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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
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
Second-order Eulerian triangle T(n,k), 1 <= k <= n.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Second-order Eulerian triangle T(n,k), 1 <= k <= n.?
Related ?
Same fingerprint ?
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.