A327997 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(-54 + 45\,n - 9\,n^{2}\right)\,P_{n-4}(x) + \left(-6 - 24\,x - 9\,n + 12\,n x + 6\,n^{2}\right)\,P_{n-3}(x) + \left(24\,x^{2} - 12\,n x^{2}\right)\,P_{n-3}'(x) + \left(8 - 4\,x + 9\,x^{2} - 8\,n - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(4\,x^{2} + 4\,x^{3} + 4\,n x^{2}\right)\,P_{n-2}'(x) + \left(3 + 4\,x + 2\,n\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)
\(P_{2}(x) = 9 + 28\,x + 9\,x^{2}\)
\(P_{3}(x) = 27 + 152\,x + 108\,x^{2} + 16\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 3 | 4 | 7 | |||||||||||||
| 2 | 9 | 28 | 9 | 46 | ||||||||||||
| 3 | 27 | 152 | 108 | 16 | 303 | |||||||||||
| 4 | 81 | 768 | 909 | 288 | 25 | 2071 | ||||||||||
| 5 | 243 | 3876 | 6795 | 3440 | 625 | 36 | 15015 | |||||||||
| 6 | 729 | 20460 | 49446 | 35280 | 10000 | 1188 | 49 | 117152 | ||||||||
| 7 | 2187 | 117288 | 368739 | 343504 | 133875 | 24444 | 2058 | 64 | 992159 | |||||||
| 8 | 6561 | 751968 | 2912922 | 3349696 | 1665475 | 412272 | 52822 | 3328 | 81 | 9155125 | ||||||
| 9 | 19683 | 5482980 | 24884334 | 33781088 | 20341125 | 6346620 | 1092798 | 104064 | 5103 | 100 | 92057895 | |||||
| 10 | 59049 | 45571356 | 232299819 | 359170080 | 251908625 | 94319316 | 20325837 | 2586240 | 190755 | 7500 | 121 | 1006438698 | ||||
| 11 | 177147 | 428174424 | 2375903178 | 4068755856 | 3225781625 | 1398500928 | 359831010 | 57047232 | 5599935 | 330000 | 10648 | 144 | 11920112127 | |||
| 12 | 531441 | 4491295776 | 26560142919 | 49333797216 | 43205389150 | 21111601104 | 6270396671 | 1183298688 | 144304578 | 11286000 | 544379 | 14688 | 169 | 152312602779 | ||
| 13 | 1594323 | 51957283140 | 322975044873 | 640749779280 | 609036756900 | 328543743528 | 109837268541 | 23885685824 | 3460583412 | 335563800 | 21438417 | 862992 | 19773 | 196 | 2090805624999 | |
| 14 | 4782969 | 656162372268 | 4248659031192 | 8902156234608 | 9060749463675 | 5310360383328 | 1960035366289 | 479483246528 | 79971777171 | 9201192000 | 727816089 | 38748528 | 1322594 | 26068 | 225 | 30707551763532 |
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Technical details
- Node id
n-9017368722c57371· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-c3bf6a1d4d14368e
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
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 3, 4; 2: 9, 28, 9; 3: 27, 152, 108, 16; 4: 81, 768, 909, 288, 25; 5: 243, 3876, 6795, 3440, 625, 36; 6: 729, 20460, 49446, 35280, 10000, 1188, 49; 7: 2187, 117288, 368739, 343504, 133875, 24444, 2058, 64; 8: 6561, 751968, 2912922, 3349696, 1665475, 412272, 52822, 3328, 81; 9: 19683, 5482980, 24884334, 33781088, 20341125, 6346620, 1092798, 104064, 5103, 100;
%F FormulaThe row polynomials satisfy P(n,x) = -(9*n^2-45*n+54)*P(n-4,x) + (6*n^2-9*n-6+(12*n-24)*x)*P(n-3,x) - (12*n-24)*x^2*d/dx P(n-3,x) - (n^2+8*n-8+(4*n+4)*x-9*x^2)*P(n-2,x) + ((4*n+4)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n+3+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 9 + 28*x + 9*x^2, P(3,x) = 27 + 152*x + 108*x^2 + 16*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[2, x_] := 9 + 28*x + 9*x^2;
P[3, x_] := 27 + 152*x + 108*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 45*n + 54)*P[n - 4, x] + (6*n^2 - 9*n - 6 + (12*n - 24)*x)*P[n - 3, x] - (12*n - 24)*x^2*D[P[n - 3, x], x] - (n^2 + 8*n - 8 + (4*n + 4)*x - 9*x^2)*P[n - 2, x] + ((4*n + 4)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n + 3 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], 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] = 3 + 4*x; v[3] = 9 + 28*x + 9*x^2; v[4] = 27 + 152*x + 108*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(9*n^2 - 45*n + 54)*v[i - 4] + (6*n^2 - 9*n - 6 + (12*n - 24)*x)*v[i - 3] - (12*n - 24)*x^2*deriv(v[i - 3], x) - (n^2 + 8*n - 8 + (4*n + 4)*x - 9*x^2)*v[i - 2] + ((4*n + 4)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n + 3 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], 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)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [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.
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023