A120986 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(2\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(4\,x + 2\,x^{2} + 2\,n - 26\,n x - 3\,n x^{2} + 4\,n^{2} + 22\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x + x^{2} + x^{3} + 8\,n x - 7\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 2 | 3 | ||||||
| 3 | 1 | 6 | 5 | 12 | |||||
| 4 | 1 | 12 | 28 | 14 | 55 | ||||
| 5 | 1 | 20 | 90 | 120 | 42 | 273 | |||
| 6 | 1 | 30 | 220 | 550 | 495 | 132 | 1428 | ||
| 7 | 1 | 42 | 455 | 1820 | 3003 | 2002 | 429 | 7752 | |
| 8 | 1 | 56 | 840 | 4900 | 12740 | 15288 | 8008 | 1430 | 43263 |
Compute more
?
Technical details
- Node id
n-849696e2738fcb3d· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-85c07d3149a3188a
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, 6, 5; 4: 1, 12, 28, 14; 5: 1, 20, 90, 120, 42; 6: 1, 30, 220, 550, 495, 132; 7: 1, 42, 455, 1820, 3003, 2002, 429; 8: 1, 56, 840, 4900, 12740, 15288, 8008, 1430; 9: 1, 72, 1428, 11424, 42840, 79968, 74256, 31824, 4862; 10: 1, 90, 2280, 23940, 122094, 325584, 465120, 348840, 125970, 16796;
%F FormulaThe row polynomials satisfy (4*n^2+2*n)*P(n,x) = (4*n^2+2*n+(22*n^2-26*n+4)*x+(n^2-3*n+2)*x^2)*P(n-1,x) + ((8*n-2)*x-(7*n-1)*x^2-(n-1)*x^3)*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[((4*n^2 + 2*n + (22*n^2 - 26*n + 4)*x + (n^2 - 3*n + 2)*x^2)*P[n - 1, x] + ((8*n - 2)*x - (7*n - 1)*x^2 - (n - 1)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 2*n)]];
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] = ((4*n^2 + 2*n + (22*n^2 - 26*n + 4)*x + (n^2 - 3*n + 2)*x^2)*v[i - 1] + ((8*n - 2)*x - (7*n - 1)*x^2 - (n - 1)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 2*n)); 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 (−∞, 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) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).?
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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) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).?
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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.