A249248 → reverse → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(1 + n\right)\,P_{n}(x) = \left(-18\,n x^{2} + 9\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-4}(x) + \left(-8\,n x^{3} + 4\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-4}'(x) + \left(-8\,n x + 27\,n x^{2} - 4\,n^{2} x + 18\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(8\,n x^{2} + 12\,n x^{3} + 4\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-3}'(x) + \left(-2 - 4\,x + 9\,x^{2} - n + 4\,n x + 9\,n x^{2} + 4\,n^{2} x\right)\,P_{n-2}(x) + \left(4\,x^{2} + 4\,x^{3} - 4\,n x^{2} + 4\,n x^{3} - 4\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(3 + 4\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)
\(P_{2}(x) = 3 + 28\,x + 9\,x^{2}\)
\(P_{3}(x) = 3 + 88\,x + 108\,x^{2} + 16\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 4 | 7 | ||||||
| 2 | 3 | 28 | 9 | 40 | |||||
| 3 | 3 | 88 | 108 | 16 | 215 | ||||
| 4 | 3 | 160 | 621 | 288 | 25 | 1097 | |||
| 5 | 3 | 244 | 2169 | 2544 | 625 | 36 | 5621 | ||
| 6 | 3 | 340 | 5238 | 14128 | 7750 | 1188 | 49 | 28696 | |
| 7 | 3 | 448 | 10395 | 54288 | 61625 | 19548 | 2058 | 64 | 148429 |
Compute more
?
Technical details
- Node id
n-9b509285184957e1· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-46a2daf31cb6550f
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: 3, 28, 9; 3: 3, 88, 108, 16; 4: 3, 160, 621, 288, 25; 5: 3, 244, 2169, 2544, 625, 36; 6: 3, 340, 5238, 14128, 7750, 1188, 49; 7: 3, 448, 10395, 54288, 61625, 19548, 2058, 64; 8: 3, 568, 18288, 156576, 345125, 208728, 43218, 3328, 81; 9: 3, 700, 29646, 373888, 1438025, 1593108, 593390, 86656, 5103, 100;
%F FormulaThe row polynomials satisfy (n+1)*P(n,x) = (9*n^3+9*n^2-18*n)*x^2*P(n-4,x) + (4*n^3+4*n^2-8*n)*x^3*d/dx P(n-4,x) - ((4*n^2+8*n)*x-(18*n^2+27*n)*x^2)*P(n-3,x) + ((4*n^2+8*n)*x^2+(8*n^2+12*n)*x^3)*d/dx P(n-3,x) - (n+2-(4*n^2+4*n-4)*x-(9*n+9)*x^2)*P(n-2,x) - ((4*n^2+4*n-4)*x^2-(4*n+4)*x^3)*d/dx P(n-2,x) + (2*n+3+(4*n+4)*x)*P(n-1,x) - (4*n+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) = 3 + 28*x + 9*x^2, P(3,x) = 3 + 88*x + 108*x^2 + 16*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[2, x_] := 3 + 28*x + 9*x^2;
P[3, x_] := 3 + 88*x + 108*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^3 + 9*n^2 - 18*n)*x^2*P[n - 4, x] + (4*n^3 + 4*n^2 - 8*n)*x^3*D[P[n - 4, x], x] - ((4*n^2 + 8*n)*x - (18*n^2 + 27*n)*x^2)*P[n - 3, x] + ((4*n^2 + 8*n)*x^2 + (8*n^2 + 12*n)*x^3)*D[P[n - 3, x], x] - (n + 2 - (4*n^2 + 4*n - 4)*x - (9*n + 9)*x^2)*P[n - 2, x] - ((4*n^2 + 4*n - 4)*x^2 - (4*n + 4)*x^3)*D[P[n - 2, x], x] + (2*n + 3 + (4*n + 4)*x)*P[n - 1, x] - (4*n + 4)*x^2*D[P[n - 1, x], x])/(n + 1)]];
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] = 3 + 28*x + 9*x^2; v[4] = 3 + 88*x + 108*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((9*n^3 + 9*n^2 - 18*n)*x^2*v[i - 4] + (4*n^3 + 4*n^2 - 8*n)*x^3*deriv(v[i - 4], x) - ((4*n^2 + 8*n)*x - (18*n^2 + 27*n)*x^2)*v[i - 3] + ((4*n^2 + 8*n)*x^2 + (8*n^2 + 12*n)*x^3)*deriv(v[i - 3], x) - (n + 2 - (4*n^2 + 4*n - 4)*x - (9*n + 9)*x^2)*v[i - 2] - ((4*n^2 + 4*n - 4)*x^2 - (4*n + 4)*x^3)*deriv(v[i - 2], x) + (2*n + 3 + (4*n + 4)*x)*v[i - 1] - (4*n + 4)*x^2*deriv(v[i - 1], x))/(n + 1)); 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]
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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
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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
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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.