A163626 → x ↦ x+1
Proved for all n all roots real, in (−1, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(-x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 0 | -1 | -1 | ||||||
| 3 | 0 | 1 | 2 | 3 | |||||
| 4 | 0 | -1 | -6 | -6 | -13 | ||||
| 5 | 0 | 1 | 14 | 36 | 24 | 75 | |||
| 6 | 0 | -1 | -30 | -150 | -240 | -120 | -541 | ||
| 7 | 0 | 1 | 62 | 540 | 1560 | 1800 | 720 | 4683 | |
| 8 | 0 | -1 | -126 | -1806 | -8400 | -16800 | -15120 | -5040 | -47293 |
Compute more
?
Technical details
- Node id
n-8ae1ed4c0b89ddf6· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-a83eeccfb5efd735
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): no - 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: 0, -1; 3: 0, 1, 2; 4: 0, -1, -6, -6; 5: 0, 1, 14, 36, 24; 6: 0, -1, -30, -150, -240, -120; 7: 0, 1, 62, 540, 1560, 1800, 720; 8: 0, -1, -126, -1806, -8400, -16800, -15120, -5040; 9: 0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320; 10: 0, -1, -510, -18150, -186480, -834120, -1905120, -2328480, -1451520, -362880;
%F FormulaThe row polynomials satisfy P(n,x) = -x*P(n-1,x) - (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[-x*P[n - 1, x] - (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] = -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, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (-1, 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)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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from
Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).?
Related ?
Same fingerprint ?
- A257620 → x ↦ x/(x+1)
- A131689: Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.
- row-factor cofactor of A101842 → x ↦ x/(x+1)
- A028246 → θ+1
- A008292 → x ↦ x+1
- A090582: T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1…
- A257620 → x ↦ x+1
- A173018 → reverse (shift +1) → x ↦ x/(x+1)
- A019538: Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
- A173018 → x ↦ x+1
- A173018 → reverse (shift +1) → reverse → x ↦ x+1
- A257609 → x ↦ x/(x+1)
- A257609 → x ↦ x+1
References
- [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741