A028246 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0)coefficients of one signHadamard kernel
Recurrence
\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 5\,x + 9\,x^{2} + 7\,x^{3} + 2\,x^{4} - n x - 4\,n x^{2} - 5\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}(x) + \left(x + 5\,x^{2} + 9\,x^{3} + 7\,x^{4} + 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 1 | 5 | 6 | 12 | |||||
| 3 | 1 | 10 | 29 | 26 | 66 | ||||
| 4 | 1 | 19 | 101 | 209 | 150 | 480 | |||
| 5 | 1 | 36 | 314 | 1126 | 1809 | 1082 | 4368 | ||
| 6 | 1 | 69 | 932 | 5158 | 13917 | 18269 | 9366 | 47712 | |
| 7 | 1 | 134 | 2715 | 21806 | 87919 | 190002 | 210853 | 94586 | 608016 |
Compute more
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Technical details
- Node id
n-3fa0e81aa186d9ff· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-179ce8abeeb2cd96
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: 1, 2; 2: 1, 5, 6; 3: 1, 10, 29, 26; 4: 1, 19, 101, 209, 150; 5: 1, 36, 314, 1126, 1809, 1082; 6: 1, 69, 932, 5158, 13917, 18269, 9366; 7: 1, 134, 2715, 21806, 87919, 190002, 210853, 94586; 8: 1, 263, 7863, 88331, 499669, 1580997, 2848477, 2737769, 1091670; 9: 1, 520, 22784, 349642, 2668982, 11600686, 30170960, 46597894, 39497657, 14174522;
%F FormulaThe row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(n-5)*x-(4*n-9)*x^2-(5*n-7)*x^3-(2*n-2)*x^4)*P(n-1,x) + (x+5*x^2+9*x^3+7*x^4+2*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 - (n - 5)*x - (4*n - 9)*x^2 - (5*n - 7)*x^3 - (2*n - 2)*x^4)*P[n - 1, x] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 - (n - 5)*x - (4*n - 9)*x^2 - (5*n - 7)*x^3 - (2*n - 2)*x^4)*v[i - 1] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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 [-1, 0)
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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
Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.?
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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
Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.?
Related ?
Same fingerprint ?
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