A019538 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0)coefficients of one signHadamard kernel
Recurrence
\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 7\,x + 15\,x^{2} + 13\,x^{3} + 4\,x^{4} - n x - 4\,n x^{2} - 5\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}(x) + \left(x + 5\,x^{2} + 9\,x^{3} + 7\,x^{4} + 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 3 | 4 | ||||||
| 3 | 1 | 8 | 13 | 22 | |||||
| 4 | 1 | 17 | 67 | 75 | 160 | ||||
| 5 | 1 | 34 | 246 | 634 | 541 | 1456 | |||
| 6 | 1 | 67 | 798 | 3562 | 6793 | 4683 | 15904 | ||
| 7 | 1 | 132 | 2451 | 16904 | 54111 | 81780 | 47293 | 202672 | |
| 8 | 1 | 261 | 7341 | 73649 | 352371 | 876255 | 1095967 | 545835 | 2951680 |
Compute more
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Technical details
- Node id
n-6042ba4cd721015a· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-6c43fbf0495b302d
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): 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; 3: 1, 8, 13; 4: 1, 17, 67, 75; 5: 1, 34, 246, 634, 541; 6: 1, 67, 798, 3562, 6793, 4683; 7: 1, 132, 2451, 16904, 54111, 81780, 47293; 8: 1, 261, 7341, 73649, 352371, 876255, 1095967, 545835; 9: 1, 518, 21748, 306146, 2056690, 7487306, 15196348, 16205198, 7087261; 10: 1, 1031, 64192, 1239080, 11247058, 56259206, 164793976, 282270272, 262283125, 102247563;
%F FormulaThe row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(n-7)*x-(4*n-15)*x^2-(5*n-13)*x^3-(2*n-4)*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 >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 - (n - 7)*x - (4*n - 15)*x^2 - (5*n - 13)*x^3 - (2*n - 4)*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, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 - (n - 7)*x - (4*n - 15)*x^2 - (5*n - 13)*x^3 - (2*n - 4)*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, 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
Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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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 of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).?
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