A281000 → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-2 - 3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2 + 8\,x - 16\,n x + 8\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-12\,x + 3\,x^{2} + 16\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 1 | 0 | |||||||
| 2 | 1 | 1 | ||||||
| 3 | 10 | 2 | 12 | |||||
| 4 | 70 | 42 | 3 | 115 | ||||
| 5 | 420 | 504 | 108 | 4 | 1036 | |||
| 6 | 2310 | 4620 | 1980 | 220 | 5 | 9135 | ||
| 7 | 12012 | 36036 | 25740 | 5720 | 390 | 6 | 79904 | |
| 8 | 60060 | 252252 | 270270 | 100100 | 13650 | 630 | 7 | 696969 |
Compute more
?
Technical details
- Node id
n-543256a15ee14711· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-e65c25f9531a60c5
Observed on the stored rows ?
- Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
- 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: 0; 2: 1; 3: 10, 2; 4: 70, 42, 3; 5: 420, 504, 108, 4; 6: 2310, 4620, 1980, 220, 5; 7: 12012, 36036, 25740, 5720, 390, 6; 8: 60060, 252252, 270270, 100100, 13650, 630, 7; 9: 291720, 1633632, 2450448, 1361360, 309400, 28560, 952, 8; 10: 1385670, 9976824, 19953648, 15519504, 5290740, 813960, 54264, 1368, 9;
%F FormulaThe row polynomials satisfy (2*n^2-3*n-2)*P(n,x) = (8*n^2-2+(6*n^2-16*n+8)*x)*P(n-1,x) + ((16*n-12)*x-(4*n-3)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t MathematicaP[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 2 + (6*n^2 - 16*n + 8)*x)*P[n - 1, x] + ((16*n - 12)*x - (4*n - 3)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 3*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 2 + (6*n^2 - 16*n + 8)*x)*v[i - 1] + ((16*n - 12)*x - (4*n - 3)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 3*n - 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 (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(2*n+1, 2*k+1)*binomial(2*n-2*k, n-k)/(n+1-k) for 0 <= k <= 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) = binomial(2*n+1, 2*k+1)*binomial(2*n-2*k, n-k)/(n+1-k) for 0 <= k <= n.?
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.