A056242 → reverse → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-14 - 20\,x + 27\,n + 38\,n x - 16\,n^{2} - 22\,n^{2} x + 3\,n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(-16 - 56\,x - 70\,x^{2} + 36\,n + 121\,n x + 116\,n x^{2} - 26\,n^{2} - 82\,n^{2} x - 66\,n^{2} x^{2} + 6\,n^{3} + 17\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x - 28\,x^{2} - 20\,x^{3} + 14\,n x + 38\,n x^{2} + 24\,n x^{3} - 6\,n^{2} x - 14\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 1 | 0 | |||||||
| 2 | 1 | 1 | ||||||
| 3 | 5 | 2 | 7 | |||||
| 4 | 16 | 18 | 3 | 37 | ||||
| 5 | 44 | 82 | 42 | 4 | 172 | |||
| 6 | 112 | 292 | 255 | 80 | 5 | 744 | ||
| 7 | 272 | 912 | 1131 | 620 | 135 | 6 | 3076 | |
| 8 | 640 | 2624 | 4224 | 3332 | 1295 | 210 | 7 | 12332 |
Compute more
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Technical details
- Node id
n-7b839014ea990f5b· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-e20dbbeeb0459415
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: 5, 2; 4: 16, 18, 3; 5: 44, 82, 42, 4; 6: 112, 292, 255, 80, 5; 7: 272, 912, 1131, 620, 135, 6; 8: 640, 2624, 4224, 3332, 1295, 210, 7; 9: 1472, 7136, 14136, 14596, 8260, 2436, 308, 8; 10: 3328, 18624, 43824, 56008, 41805, 18144, 4242, 432, 9;
%F FormulaThe row polynomials satisfy (3*n^3-16*n^2+27*n-14+(4*n^3-22*n^2+38*n-20)*x)*P(n,x) = (6*n^3-26*n^2+36*n-16+(17*n^3-82*n^2+121*n-56)*x+(12*n^3-66*n^2+116*n-70)*x^2)*P(n-1,x) - ((6*n^2-14*n+8)*x+(14*n^2-38*n+28)*x^2+(8*n^2-24*n+20)*x^3)*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[((6*n^3 - 26*n^2 + 36*n - 16 + (17*n^3 - 82*n^2 + 121*n - 56)*x + (12*n^3 - 66*n^2 + 116*n - 70)*x^2)*P[n - 1, x] - ((6*n^2 - 14*n + 8)*x + (14*n^2 - 38*n + 28)*x^2 + (8*n^2 - 24*n + 20)*x^3)*D[P[n - 1, x], x])/(3*n^3 - 16*n^2 + 27*n - 14 + (4*n^3 - 22*n^2 + 38*n - 20)*x)]];
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] = ((6*n^3 - 26*n^2 + 36*n - 16 + (17*n^3 - 82*n^2 + 121*n - 56)*x + (12*n^3 - 66*n^2 + 116*n - 70)*x^2)*v[i - 1] - ((6*n^2 - 14*n + 8)*x + (14*n^2 - 38*n + 28)*x^2 + (8*n^2 - 24*n + 20)*x^3)*deriv(v[i - 1], x))/(3*n^3 - 16*n^2 + 27*n - 14 + (4*n^3 - 22*n^2 + 38*n - 20)*x)); 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
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
no recorded property covers this requirement
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from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
References
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.