A081582 → γ → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(n + 7\,n^{2} x\right)\,P_{n}(x) = \left(-28\,x - 196\,x^{2} + n + 28\,n x + 98\,n x^{2} + 7\,n^{2} x + 98\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-x - 42\,x^{2} - 392\,x^{3} - 7\,n x^{2} - 196\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 1 | 1 | |||
| 2 | 1 | 14 | 15 | ||
| 3 | 1 | 28 | 29 | ||
| 4 | 1 | 42 | 147 | 190 | |
| 5 | 1 | 56 | 441 | 498 | |
| 6 | 1 | 70 | 882 | 1372 | 2325 |
| 7 | 1 | 84 | 1470 | 5488 | 7043 |
Compute more
?
Technical details
- Node id
n-73c8436fb109f7ec· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-3dfc77d38090b9d0
Observed on the stored rows ?
- Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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: 1, 14; 3: 1, 28; 4: 1, 42, 147; 5: 1, 56, 441; 6: 1, 70, 882, 1372; 7: 1, 84, 1470, 5488; 8: 1, 98, 2205, 13720, 12005; 9: 1, 112, 3087, 27440, 60025;
%F FormulaThe row polynomials satisfy (n+7*n^2*x)*P(n,x) = (n+(7*n^2+28*n-28)*x+(98*n^2+98*n-196)*x^2)*P(n-1,x) - (x+(7*n+42)*x^2+(196*n+392)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + (7*n^2 + 28*n - 28)*x + (98*n^2 + 98*n - 196)*x^2)*P[n - 1, x] - (x + (7*n + 42)*x^2 + (196*n + 392)*x^3)*D[P[n - 1, x], x])/(n + 7*n^2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + (7*n^2 + 28*n - 28)*x + (98*n^2 + 98*n - 196)*x^2)*v[i - 1] - (x + (7*n + 42)*x^2 + (196*n + 392)*x^3)*deriv(v[i - 1], x))/(n + 7*n^2*x)); 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]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
gamma_polynomial of Pascal-(1,7,1) array.?
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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from
Pascal-(1,7,1) array.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
Pascal-(1,7,1) array.?
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from
gamma_polynomial of Pascal-(1,7,1) array.?
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.