A062190 → 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
\(\left(-5\,n - n^{2} + 25\,n^{2} x + 10\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(-40\,x + 100\,x^{2} - 5\,n - 26\,n x + 185\,n x^{2} - n^{2} + 19\,n^{2} x + 112\,n^{2} x^{2} + 10\,n^{3} x + 32\,n^{3} x^{2} + n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-5\,x + 30\,x^{2} - 25\,x^{3} - 2\,n x + 42\,n x^{2} - 40\,n x^{3} + 18\,n^{2} x^{2} - 18\,n^{2} x^{3} + 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 12 | 13 | ||||||
| 2 | 1 | 28 | 63 | 92 | |||||
| 3 | 1 | 48 | 252 | 224 | 525 | ||||
| 4 | 1 | 72 | 648 | 1344 | 630 | 2695 | |||
| 5 | 1 | 100 | 1350 | 4800 | 5250 | 1512 | 13013 | ||
| 6 | 1 | 132 | 2475 | 13200 | 24750 | 16632 | 3234 | 60424 | |
| 7 | 1 | 168 | 4158 | 30800 | 86625 | 99792 | 45276 | 6336 | 273156 |
Computed in your browser ?
Technical details
- Node id
n-27ff36c5792a7014· revision 1 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-c1b99877cd304482
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: 1, 12; 2: 1, 28, 63; 3: 1, 48, 252, 224; 4: 1, 72, 648, 1344, 630; 5: 1, 100, 1350, 4800, 5250, 1512; 6: 1, 132, 2475, 13200, 24750, 16632, 3234; 7: 1, 168, 4158, 30800, 86625, 99792, 45276, 6336; 8: 1, 208, 6552, 64064, 250250, 432432, 336336, 109824, 11583; 9: 1, 252, 9828, 122304, 630630, 1513512, 1765764, 988416, 243243, 20020;
%F FormulaThe row polynomials satisfy -(n^2+5*n-(n^4+10*n^3+25*n^2)*x)*P(n,x) = -(n^2+5*n-(n^4+10*n^3+19*n^2-26*n-40)*x-(3*n^4+32*n^3+112*n^2+185*n+100)*x^2)*P(n-1,x) - ((2*n+5)*x-(2*n^3+18*n^2+42*n+30)*x^2+(2*n^3+18*n^2+40*n+25)*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^2 + 5*n - (n^4 + 10*n^3 + 19*n^2 - 26*n - 40)*x - (3*n^4 + 32*n^3 + 112*n^2 + 185*n + 100)*x^2)*P[n - 1, x] - ((2*n + 5)*x - (2*n^3 + 18*n^2 + 42*n + 30)*x^2 + (2*n^3 + 18*n^2 + 40*n + 25)*x^3)*D[P[n - 1, x], x])/(-n^2 - 5*n + (n^4 + 10*n^3 + 25*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^2 + 5*n - (n^4 + 10*n^3 + 19*n^2 - 26*n - 40)*x - (3*n^4 + 32*n^3 + 112*n^2 + 185*n + 100)*x^2)*v[i - 1] - ((2*n + 5)*x - (2*n^3 + 18*n^2 + 42*n + 30)*x^2 + (2*n^3 + 18*n^2 + 40*n + 25)*x^3)*deriv(v[i - 1], x))/(-n^2 - 5*n + (n^4 + 10*n^3 + 25*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)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-595b0519ab8ba235 -
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-595b0519ab8ba235 -
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-595b0519ab8ba235 -
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-595b0519ab8ba235
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-595b0519ab8ba235Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
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premise (sequence):
Coefficient triangle of certain polynomials N(5; m,x).
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-e968c429d4b85e0broot: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
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premise (sequence):
Coefficient triangle of certain polynomials N(5; m,x).
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.