A113413 → θ+1
nonnegative coefficients
Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means
Recurrence
\(\left(n + n x - n x^{2} - n x^{3} - n^{2} + 5\,n^{2} x + 5\,n^{2} x^{2} - n^{2} x^{3} - 2\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-8\,x^{2} - 6\,x^{3} - 2\,x^{4} + n + 13\,n x + 23\,n x^{2} + 5\,n x^{3} - 2\,n x^{4} - n^{2} - 3\,n^{2} x + 11\,n^{2} x^{2} + 9\,n^{2} x^{3} - 2\,n^{3} x - 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-x^{2} - 7\,x^{3} - 7\,x^{4} - x^{5} + n x + 8\,n x^{2} + 12\,n x^{3} - 4\,n x^{4} - n x^{5} + 2\,n^{2} x^{2} + 12\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 2 | 4 | ||||||
| 2 | 2 | 8 | 3 | 13 | |||||
| 3 | 2 | 16 | 18 | 4 | 40 | ||||
| 4 | 2 | 24 | 54 | 32 | 5 | 117 | |||
| 5 | 2 | 32 | 114 | 128 | 50 | 6 | 332 | ||
| 6 | 2 | 40 | 198 | 352 | 250 | 72 | 7 | 921 | |
| 7 | 2 | 48 | 306 | 768 | 850 | 432 | 98 | 8 | 2512 |
Computed in your browser ?
Technical details
- Node id
n-ef1f2f213ef4dbf2· revision 1 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-814a2de3509dad73
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 ?
- A119915: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
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, 2; 2: 2, 8, 3; 3: 2, 16, 18, 4; 4: 2, 24, 54, 32, 5; 5: 2, 32, 114, 128, 50, 6; 6: 2, 40, 198, 352, 250, 72, 7; 7: 2, 48, 306, 768, 850, 432, 98, 8; 8: 2, 56, 438, 1440, 2250, 1752, 686, 128, 9; 9: 2, 64, 594, 2432, 5010, 5472, 3234, 1024, 162, 10;
%F FormulaThe row polynomials satisfy -(n^2-n+(2*n^3-5*n^2-n)*x-(2*n^3+5*n^2-n)*x^2+(n^2+n)*x^3)*P(n,x) = -(n^2-n+(2*n^3+3*n^2-13*n)*x+(6*n^3-11*n^2-23*n+8)*x^2-(9*n^2+5*n-6)*x^3+(2*n+2)*x^4)*P(n-1,x) + (n*x+(2*n^2+8*n-1)*x^2+(12*n^2+12*n-7)*x^3+(2*n^2-4*n-7)*x^4-(n+1)*x^5)*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^2 - n + (2*n^3 + 3*n^2 - 13*n)*x + (6*n^3 - 11*n^2 - 23*n + 8)*x^2 - (9*n^2 + 5*n - 6)*x^3 + (2*n + 2)*x^4)*P[n - 1, x] + (n*x + (2*n^2 + 8*n - 1)*x^2 + (12*n^2 + 12*n - 7)*x^3 + (2*n^2 - 4*n - 7)*x^4 - (n + 1)*x^5)*D[P[n - 1, x], x])/(-n^2 + n - (2*n^3 - 5*n^2 - n)*x + (2*n^3 + 5*n^2 - n)*x^2 - (n^2 + n)*x^3)]];
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^2 - n + (2*n^3 + 3*n^2 - 13*n)*x + (6*n^3 - 11*n^2 - 23*n + 8)*x^2 - (9*n^2 + 5*n - 6)*x^3 + (2*n + 2)*x^4)*v[i - 1] + (n*x + (2*n^2 + 8*n - 1)*x^2 + (12*n^2 + 12*n - 7)*x^3 + (2*n^2 - 4*n - 7)*x^4 - (n + 1)*x^5)*deriv(v[i - 1], x))/(-n^2 + n - (2*n^3 - 5*n^2 - n)*x + (2*n^3 + 5*n^2 - n)*x^2 - (n^2 + n)*x^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)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-ef030cea3fd1d83e -
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-ef030cea3fd1d83e -
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-ef030cea3fd1d83e -
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-ef030cea3fd1d83e
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 euler_polynomial, shift 0
e-ef030cea3fd1d83eLaguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
-
premise (sequence):
A Riordan array of coordination sequences.
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_derivative_lag
e-7fe1799d89540b32root: theorem checkLiu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
-
-
premise (sequence):
A Riordan array of coordination sequences.
Related sequences ?
Same fingerprint (1)
Equal up to content, sign, reversal and a small index shift.
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.