A039756 → D+1
nonnegative coefficients
Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means
Recurrence
\(\left(-3 + 4\,x + 2\,n\right)\,P_{n}(x) = \left(1 - 14\,x + 57\,x^{2} - 74\,x^{3} + 24\,x^{4} - 2\,n + 24\,n x - 66\,n x^{2} + 80\,n x^{3} - 16\,n x^{4} - 8\,n^{2} x + 36\,n^{2} x^{2} - 24\,n^{2} x^{3} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} + 14\,x^{3} - 28\,x^{4} + 16\,x^{5} + 4\,n x^{2} - 16\,n x^{3} + 24\,n x^{4} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-4 + 26\,x - 34\,x^{2} + 8\,x^{3} + 4\,n - 24\,n x + 20\,n x^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(6\,x^{2} - 8\,x^{3} - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 5 | 6 | 1 | 12 | |||||
| 3 | 10 | 35 | 16 | 1 | 62 | ||||
| 4 | 17 | 132 | 178 | 44 | 1 | 372 | |||
| 5 | 26 | 365 | 1160 | 814 | 126 | 1 | 2492 | ||
| 6 | 37 | 826 | 4955 | 8604 | 3591 | 370 | 1 | 18384 | |
| 7 | 50 | 1631 | 16016 | 54719 | 58506 | 15777 | 1100 | 1 | 147800 |
Computed in your browser ?
Technical details
- Node id
n-61170c61db6a737a· revision 1 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-d465d755c34dd8dd
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
OEIS matches ?
No OEIS matches.
OEIS submission text ?
Fields to paste into an OEIS submission: the example %e, the formula
%F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows.
For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched;
the matched terms were compared with the published ones, which identifies the node with the entry on a
finite prefix, not for every n. The PARI/GP program is not machine-checked here.
The atlas's own rows: n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 2, 1; 2: 5, 6, 1; 3: 10, 35, 16, 1; 4: 17, 132, 178, 44, 1; 5: 26, 365, 1160, 814, 126, 1; 6: 37, 826, 4955, 8604, 3591, 370, 1; 7: 50, 1631, 16016, 54719, 58506, 15777, 1100, 1; 8: 65, 2920, 42756, 246120, 538566, 379224, 70148, 3288, 1; 9: 82, 4857, 99408, 870828, 3303972, 4938030, 2394400, 317948, 9850, 1;
%F FormulaThe row polynomials satisfy (2*n-3+4*x)*P(n,x) = -(2*n-1+(8*n^2-24*n+14)*x+(8*n^3-36*n^2+66*n-57)*x^2+(24*n^2-80*n+74)*x^3+(16*n-24)*x^4)*P(n-2,x) + ((4*n-2)*x^2+(8*n^2-16*n+14)*x^3+(24*n-28)*x^4+16*x^5)*d/dx P(n-2,x) + (4*n-4+(8*n^2-24*n+26)*x+(20*n-34)*x^2+8*x^3)*P(n-1,x) - ((4*n-6)*x^2+8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 + (8*n^2 - 24*n + 14)*x + (8*n^3 - 36*n^2 + 66*n - 57)*x^2 + (24*n^2 - 80*n + 74)*x^3 + (16*n - 24)*x^4)*P[n - 2, x] + ((4*n - 2)*x^2 + (8*n^2 - 16*n + 14)*x^3 + (24*n - 28)*x^4 + 16*x^5)*D[P[n - 2, x], x] + (4*n - 4 + (8*n^2 - 24*n + 26)*x + (20*n - 34)*x^2 + 8*x^3)*P[n - 1, x] - ((4*n - 6)*x^2 + 8*x^3)*D[P[n - 1, x], x])/(2*n - 3 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 1 + (8*n^2 - 24*n + 14)*x + (8*n^3 - 36*n^2 + 66*n - 57)*x^2 + (24*n^2 - 80*n + 74)*x^3 + (16*n - 24)*x^4)*v[i - 2] + ((4*n - 2)*x^2 + (8*n^2 - 16*n + 14)*x^3 + (24*n - 28)*x^4 + 16*x^5)*deriv(v[i - 2], x) + (4*n - 4 + (8*n^2 - 24*n + 26)*x + (20*n - 34)*x^2 + 8*x^3)*v[i - 1] - ((4*n - 6)*x^2 + 8*x^3)*deriv(v[i - 1], x))/(2*n - 3 + 4*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)Generated with polytool c48629374597.
Proved properties
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Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.by the cited steps:
e-40b5448cd5018914 -
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).derived from the cited steps by a stated theorem:
e-40b5448cd5018914 -
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.derived from the cited steps by a stated theorem:
e-40b5448cd5018914 -
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.by the cited steps:
e-40b5448cd5018914
Proof chains
1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON
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preserver_image derivative_polynomial, shift 0
e-40b5448cd5018914Laguerre’s theorem on differential operators [SP14] polytool ?
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premise (sequence):
Triangle of B-analogs of Stirling numbers of 2nd kind.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
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theorem_check liu_wang
e-7519e7535554cb98root: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
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premise (sequence):
Triangle of B-analogs of Stirling numbers of 2nd kind.
Related sequences ?
No candidate equalities or related sequences.
References
- [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.