A113413 → (θ+1)²
nonnegative coefficients
Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means
Recurrence
\(\left(1 - x + x^{2} - x^{3} - 2\,n + 10\,n x + 12\,n x^{2} + 4\,n x^{3} + n^{2} - 16\,n^{2} x - 8\,n^{2} x^{2} + 2\,n^{2} x^{3} + n^{2} x^{4} + 6\,n^{3} x - 8\,n^{3} x^{2} - 6\,n^{3} x^{3} + 4\,n^{4} x^{2}\right)\,P_{n}(x) = \left(-x^{2} + x^{3} - x^{4} + x^{5} - 4\,n x^{2} - 12\,n x^{3} - 10\,n x^{4} + 2\,n x^{5} + n^{2} x + 2\,n^{2} x^{2} - 8\,n^{2} x^{3} - 16\,n^{2} x^{4} + n^{2} x^{5} + 6\,n^{3} x^{2} + 8\,n^{3} x^{3} - 6\,n^{3} x^{4} + 4\,n^{4} x^{3}\right)\,P_{n-2}(x) + \left(1 + 7\,x + 32\,x^{2} + 32\,x^{3} + 7\,x^{4} + x^{5} - 2\,n - 6\,n x - 8\,n x^{2} + 8\,n x^{3} + 6\,n x^{4} + 2\,n x^{5} + n^{2} - 9\,n^{2} x - 36\,n^{2} x^{2} - 36\,n^{2} x^{3} - 9\,n^{2} x^{4} + n^{2} x^{5} + 6\,n^{3} x + 6\,n^{3} x^{2} - 6\,n^{3} x^{3} - 6\,n^{3} x^{4} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 4 | 6 | ||||||
| 2 | 2 | 16 | 9 | 27 | |||||
| 3 | 2 | 32 | 54 | 16 | 104 | ||||
| 4 | 2 | 48 | 162 | 128 | 25 | 365 | |||
| 5 | 2 | 64 | 342 | 512 | 250 | 36 | 1206 | ||
| 6 | 2 | 80 | 594 | 1408 | 1250 | 432 | 49 | 3815 | |
| 7 | 2 | 96 | 918 | 3072 | 4250 | 2592 | 686 | 64 | 11680 |
Computed in your browser ?
Technical details
- Node id
n-ea637c45d9fe93b5· revision 1 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-3559825f68d92921
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, 4; 2: 2, 16, 9; 3: 2, 32, 54, 16; 4: 2, 48, 162, 128, 25; 5: 2, 64, 342, 512, 250, 36; 6: 2, 80, 594, 1408, 1250, 432, 49; 7: 2, 96, 918, 3072, 4250, 2592, 686, 64; 8: 2, 112, 1314, 5760, 11250, 10512, 4802, 1024, 81; 9: 2, 128, 1782, 9728, 25050, 32832, 22638, 8192, 1458, 100;
%F FormulaThe row polynomials satisfy (n^2-2*n+1+(6*n^3-16*n^2+10*n-1)*x+(4*n^4-8*n^3-8*n^2+12*n+1)*x^2-(6*n^3-2*n^2-4*n+1)*x^3+n^2*x^4)*P(n,x) = (n^2*x+(6*n^3+2*n^2-4*n-1)*x^2+(4*n^4+8*n^3-8*n^2-12*n+1)*x^3-(6*n^3+16*n^2+10*n+1)*x^4+(n^2+2*n+1)*x^5)*P(n-2,x) + (n^2-2*n+1+(6*n^3-9*n^2-6*n+7)*x+(4*n^4+6*n^3-36*n^2-8*n+32)*x^2+(4*n^4-6*n^3-36*n^2+8*n+32)*x^3-(6*n^3+9*n^2-6*n-7)*x^4+(n^2+2*n+1)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^2*x + (6*n^3 + 2*n^2 - 4*n - 1)*x^2 + (4*n^4 + 8*n^3 - 8*n^2 - 12*n + 1)*x^3 - (6*n^3 + 16*n^2 + 10*n + 1)*x^4 + (n^2 + 2*n + 1)*x^5)*P[n - 2, x] + (n^2 - 2*n + 1 + (6*n^3 - 9*n^2 - 6*n + 7)*x + (4*n^4 + 6*n^3 - 36*n^2 - 8*n + 32)*x^2 + (4*n^4 - 6*n^3 - 36*n^2 + 8*n + 32)*x^3 - (6*n^3 + 9*n^2 - 6*n - 7)*x^4 + (n^2 + 2*n + 1)*x^5)*P[n - 1, x])/(n^2 - 2*n + 1 + (6*n^3 - 16*n^2 + 10*n - 1)*x + (4*n^4 - 8*n^3 - 8*n^2 + 12*n + 1)*x^2 - (6*n^3 - 2*n^2 - 4*n + 1)*x^3 + n^2*x^4)]];
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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n^2*x + (6*n^3 + 2*n^2 - 4*n - 1)*x^2 + (4*n^4 + 8*n^3 - 8*n^2 - 12*n + 1)*x^3 - (6*n^3 + 16*n^2 + 10*n + 1)*x^4 + (n^2 + 2*n + 1)*x^5)*v[i - 2] + (n^2 - 2*n + 1 + (6*n^3 - 9*n^2 - 6*n + 7)*x + (4*n^4 + 6*n^3 - 36*n^2 - 8*n + 32)*x^2 + (4*n^4 - 6*n^3 - 36*n^2 + 8*n + 32)*x^3 - (6*n^3 + 9*n^2 - 6*n - 7)*x^4 + (n^2 + 2*n + 1)*x^5)*v[i - 1])/(n^2 - 2*n + 1 + (6*n^3 - 16*n^2 + 10*n - 1)*x + (4*n^4 - 8*n^3 - 8*n^2 + 12*n + 1)*x^2 - (6*n^3 - 2*n^2 - 4*n + 1)*x^3 + n^2*x^4)); 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-ea263f6af5fbc945 -
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-ea263f6af5fbc945 -
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-ea263f6af5fbc945 -
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-ea263f6af5fbc945
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-ea263f6af5fbc945Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
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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 ?
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premise (sequence):
A Riordan array of coordination sequences.
Related sequences ?
No candidate equalities or related sequences.
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.