basis-map coefficients of A021012 → basis A008277 → x ↦ x/(x+1)
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 5\,x + 6\,x^{2} + 2\,x^{3} - n - 6\,n x - 8\,n x^{2} - 3\,n x^{3} + n^{2} x + 2\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4} - n x - 3\,n x^{2} - 3\,n x^{3} - n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | -1 | -2 | -3 | ||||||
| 3 | 2 | 10 | 10 | 22 | |||||
| 4 | -6 | -60 | -138 | -90 | -294 | ||||
| 5 | 24 | 456 | 1824 | 2616 | 1248 | 6168 | |||
| 6 | -120 | -4320 | -26880 | -63720 | -65280 | -24360 | -184680 | ||
| 7 | 720 | 49680 | 454320 | 1586880 | 2621520 | 2070720 | 631440 | 7415280 | |
| 8 | -5040 | -675360 | -8814960 | -42693840 | -101248560 | -127048320 | -81209520 | -20865600 | -382561200 |
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Technical details
- Node id
n-36699036e6968a66· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-629d13292f83019a
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: -1, -2; 3: 2, 10, 10; 4: -6, -60, -138, -90; 5: 24, 456, 1824, 2616, 1248; 6: -120, -4320, -26880, -63720, -65280, -24360; 7: 720, 49680, 454320, 1586880, 2621520, 2070720, 631440; 8: -5040, -675360, -8814960, -42693840, -101248560, -127048320, -81209520, -20865600; 9: 40320, 10604160, 195068160, 1263265920, 4038894720, 7162081920, 7199700480, 3850358400, 852647040; 10: -362880, -188697600, -4882187520, -41298284160, -171216581760, -402605562240, -566015869440, -472633418880, -216695606400, -42085008000;
%F FormulaThe row polynomials satisfy (1+x)*P(n,x) = -(n-1-(n^2-6*n+5)*x-(2*n^2-8*n+6)*x^2-(n^2-3*n+2)*x^3)*P(n-1,x) - ((n-1)*x+(3*n-3)*x^2+(3*n-3)*x^3+(n-1)*x^4)*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[Together[(-(n - 1 - (n^2 - 6*n + 5)*x - (2*n^2 - 8*n + 6)*x^2 - (n^2 - 3*n + 2)*x^3)*P[n - 1, x] - ((n - 1)*x + (3*n - 3)*x^2 + (3*n - 3)*x^3 + (n - 1)*x^4)*D[P[n - 1, x], x])/(1 + 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] = (-(n - 1 - (n^2 - 6*n + 5)*x - (2*n^2 - 8*n + 6)*x^2 - (n^2 - 3*n + 2)*x^3)*v[i - 1] - ((n - 1)*x + (3*n - 3)*x^2 + (3*n - 3)*x^3 + (n - 1)*x^4)*deriv(v[i - 1], x))/(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)
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
basis_map_A008277 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polyno…?
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Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
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from
basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
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Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
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from
basis_map_A008277 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polyno…?
Related ?
Same fingerprint ?
- A008277 → x ↦ x/(x+1)
- A008278 → x ↦ x+1 (shift +1) → reverse
- A106800 → x ↦ x+1
- A048993 → reverse → x ↦ x+1
- A130123 → basis A008277 → x ↦ x/(x+1)
- A048993 → x ↦ x/(x+1)
- A008278 → x ↦ x+1
- A008278 → x ↦ x+1 → reverse
- basis-map coefficients of A112227 → basis A008277 → x ↦ x/(x+1)
- A059340: Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-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