A119308 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(3 + n\right)\,P_{n}(x) = \left(-2\,x + 4\,x^{2} - 2\,x^{3} + n x - 2\,n x^{2} + n x^{3}\right)\,P_{n-3}(x) + \left(x + x^{2} - n - 3\,n x^{2}\right)\,P_{n-2}(x) + \left(3 + 4\,x + 2\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
\(P_{2}(x) = 1 + 5\,x + 3\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 1 | 5 | 3 | 9 | |||||
| 3 | 1 | 9 | 14 | 4 | 28 | ||||
| 4 | 1 | 14 | 40 | 30 | 5 | 90 | |||
| 5 | 1 | 20 | 90 | 125 | 55 | 6 | 297 | ||
| 6 | 1 | 27 | 175 | 385 | 315 | 91 | 7 | 1001 | |
| 7 | 1 | 35 | 308 | 980 | 1274 | 686 | 140 | 8 | 3432 |
Compute more
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Technical details
- Node id
n-4f9b456a67845b83· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-1c718f648220d69b
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, 3; 3: 1, 9, 14, 4; 4: 1, 14, 40, 30, 5; 5: 1, 20, 90, 125, 55, 6; 6: 1, 27, 175, 385, 315, 91, 7; 7: 1, 35, 308, 980, 1274, 686, 140, 8; 8: 1, 44, 504, 2184, 4116, 3528, 1344, 204, 9; 9: 1, 54, 780, 4410, 11340, 14112, 8568, 2430, 285, 10;
%F FormulaThe row polynomials satisfy (n+3)*P(n,x) = ((n-2)*x-(2*n-4)*x^2+(n-2)*x^3)*P(n-3,x) - (n-x+(3*n-1)*x^2)*P(n-2,x) + (2*n+3+(3*n+4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 5*x + 3*x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 5*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((n - 2)*x - (2*n - 4)*x^2 + (n - 2)*x^3)*P[n - 3, x] - (n - x + (3*n - 1)*x^2)*P[n - 2, x] + (2*n + 3 + (3*n + 4)*x)*P[n - 1, x])/(n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 5*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (((n - 2)*x - (2*n - 4)*x^2 + (n - 2)*x^3)*v[i - 3] - (n - x + (3*n - 1)*x^2)*v[i - 2] + (2*n + 3 + (3*n + 4)*x)*v[i - 1])/(n + 3)); 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 (−∞, 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
Triangle for first differences of Catalan numbers.?
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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 for first differences of Catalan numbers.?
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.