A088617 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(2\,x + 12\,x^{2} + 18\,x^{3} + 8\,x^{4} + n + 3\,n x - n x^{2} - 7\,n x^{3} - 4\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 10\,x^{2} + 18\,x^{3} + 14\,x^{4} + 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 2 | 3 | ||||||
| 3 | 1 | 5 | 6 | 12 | |||||
| 4 | 1 | 9 | 25 | 22 | 57 | ||||
| 5 | 1 | 14 | 66 | 129 | 90 | 300 | |||
| 6 | 1 | 20 | 140 | 450 | 681 | 394 | 1686 | ||
| 7 | 1 | 27 | 260 | 1210 | 2955 | 3653 | 1806 | 9912 | |
| 8 | 1 | 35 | 441 | 2765 | 9625 | 18963 | 19825 | 8558 | 60213 |
Compute more
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Technical details
- Node id
n-d4b69b787803fc82· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-56b745d77a7c0967
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, 2; 3: 1, 5, 6; 4: 1, 9, 25, 22; 5: 1, 14, 66, 129, 90; 6: 1, 20, 140, 450, 681, 394; 7: 1, 27, 260, 1210, 2955, 3653, 1806; 8: 1, 35, 441, 2765, 9625, 18963, 19825, 8558; 9: 1, 44, 700, 5642, 26110, 72548, 119812, 108545, 41586; 10: 1, 54, 1056, 10584, 62244, 227808, 526512, 748548, 598417, 206098;
%F FormulaThe row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (n+(3*n+2)*x-(n-12)*x^2-(7*n-18)*x^3-(4*n-8)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*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[((n + (3*n + 2)*x - (n - 12)*x^2 - (7*n - 18)*x^3 - (4*n - 8)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*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] = ((n + (3*n + 2)*x - (n - 12)*x^2 - (7*n - 18)*x^3 - (4*n - 8)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*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 read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.?
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Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
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from
Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.?
Related ?
Same fingerprint ?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [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.