A046739 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(8\right)\,P_{n}(x) = \left(-16\,n - 20\,n x - 14\,n x^{2} + 9\,n x^{3} + 4\,n^{2} - 10\,n^{2} x - 14\,n^{2} x^{2} + 2\,n^{2} x^{3} + 12\,n^{3} + 6\,n^{3} x - 3\,n^{3} x^{3}\right)\,P_{n-4}(x) + \left(-8\,n x - 6\,n x^{2} + 9\,n x^{3} + 2\,n^{2} x - 6\,n^{2} x^{2} + 2\,n^{2} x^{3} + 6\,n^{3} x - 3\,n^{3} x^{3}\right)\,P_{n-4}'(x) + \left(24 + 6\,x + 35\,x^{2} - 25\,x^{3} - 22\,n - 16\,n x + 6\,n x^{2} - 9\,n x^{3} - 14\,n^{2} - 16\,n^{2} x - 25\,n^{2} x^{2} + 5\,n^{2} x^{3} + 12\,n^{3} + 6\,n^{3} x - 3\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(-24 + 14\,x + 41\,x^{2} - 12\,x^{3} - 9\,x^{4} + 6\,n - 34\,n x + 18\,n x^{2} + 11\,n x^{3} - 2\,n x^{4} + 18\,n^{2} - 22\,n^{2} x - 15\,n^{2} x^{2} + 8\,n^{2} x^{3} + 3\,n^{2} x^{4} + 6\,n^{3} x - 3\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(-8\,x + 2\,x^{2} + 15\,x^{3} - 9\,x^{4} + 2\,n x - 8\,n x^{2} + 8\,n x^{3} - 2\,n x^{4} + 6\,n^{2} x - 6\,n^{2} x^{2} - 3\,n^{2} x^{3} + 3\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(-12 + 84\,x - 25\,x^{2} - 14\,n + 36\,n x + 10\,n x^{2} - 16\,n x^{3} - 6\,n^{2} - 12\,n^{2} x - 5\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(8 + 20\,x - 35\,x^{2} - 4\,x^{3} + 16\,x^{4} - 2\,n + 4\,n x^{2} - 6\,n^{2} + 3\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(8\,x^{2} - 16\,x^{3} + 8\,x^{4}\right)\,P_{n-2}''(x) + \left(26 - 28\,x + 16\,x^{2} + 6\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 8 + 9\,x + x^{2}\)
\(P_{2}(x) = 22 + 63\,x + 24\,x^{2} + x^{3}\)
\(P_{3}(x) = 52 + 373\,x + 314\,x^{2} + 55\,x^{3} + x^{4}\)
\(P_{4}(x) = 114 + 1739\,x + 3252\,x^{2} + 1265\,x^{3} + 118\,x^{4} + x^{5}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 8 | 9 | 1 | 18 | |||||||
| 2 | 22 | 63 | 24 | 1 | 110 | ||||||
| 3 | 52 | 373 | 314 | 55 | 1 | 795 | |||||
| 4 | 114 | 1739 | 3252 | 1265 | 118 | 1 | 6489 | ||||
| 5 | 240 | 6961 | 26254 | 21075 | 4556 | 245 | 1 | 59332 | |||
| 6 | 494 | 25335 | 173920 | 269165 | 115938 | 15379 | 500 | 1 | 600732 | ||
| 7 | 1004 | 86733 | 1003186 | 2748143 | 2207544 | 577297 | 49886 | 1011 | 1 | 6674805 | |
| 8 | 2026 | 285123 | 5255628 | 23680761 | 32964102 | 15722175 | 2695536 | 157749 | 2034 | 1 | 80765135 |
Compute more
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Technical details
- Node id
n-614aff0891e33f9b· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-e498d6ad1a16c710
Observed on the stored rows ?
- Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
- 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 8, 9, 1; 2: 22, 63, 24, 1; 3: 52, 373, 314, 55, 1; 4: 114, 1739, 3252, 1265, 118, 1; 5: 240, 6961, 26254, 21075, 4556, 245, 1; 6: 494, 25335, 173920, 269165, 115938, 15379, 500, 1; 7: 1004, 86733, 1003186, 2748143, 2207544, 577297, 49886, 1011, 1; 8: 2026, 285123, 5255628, 23680761, 32964102, 15722175, 2695536, 157749, 2034, 1; 9: 4072, 911881, 25721942, 180119131, 405950628, 330149981, 101882866, 12053919, 490544, 4081, 1; 10: 8166, 2861615, 119850072, 1249307701, 4312602378, 5614201691, 2919237004, 618253905, 52314974, 1508375, 8176, 1;
%F FormulaThe row polynomials satisfy 8*P(n,x) = (12*n^3+4*n^2-16*n+(6*n^3-10*n^2-20*n)*x-(14*n^2+14*n)*x^2-(3*n^3-2*n^2-9*n)*x^3)*P(n-4,x) + ((6*n^3+2*n^2-8*n)*x-(6*n^2+6*n)*x^2-(3*n^3-2*n^2-9*n)*x^3)*d/dx P(n-4,x) + (12*n^3-14*n^2-22*n+24+(6*n^3-16*n^2-16*n+6)*x-(25*n^2-6*n-35)*x^2-(3*n^3-5*n^2+9*n+25)*x^3)*P(n-3,x) + (18*n^2+6*n-24+(6*n^3-22*n^2-34*n+14)*x-(15*n^2-18*n-41)*x^2-(3*n^3-8*n^2-11*n+12)*x^3+(3*n^2-2*n-9)*x^4)*d/dx P(n-3,x) + ((6*n^2+2*n-8)*x-(6*n^2+8*n-2)*x^2-(3*n^2-8*n-15)*x^3+(3*n^2-2*n-9)*x^4)*d^2/dx^2 P(n-3,x) - (6*n^2+14*n+12+(12*n^2-36*n-84)*x+(5*n^2-10*n+25)*x^2+16*n*x^3)*P(n-2,x) - (6*n^2+2*n-8-20*x-(3*n^2+4*n-35)*x^2+4*x^3-16*x^4)*d/dx P(n-2,x) + (8*x^2-16*x^3+8*x^4)*d^2/dx^2 P(n-2,x) + (6*n+26+(16*n-28)*x+16*x^2)*P(n-1,x) for n >= 5, with P(1,x) = 8 + 9*x + x^2, P(2,x) = 22 + 63*x + 24*x^2 + x^3, P(3,x) = 52 + 373*x + 314*x^2 + 55*x^3 + x^4, P(4,x) = 114 + 1739*x + 3252*x^2 + 1265*x^3 + 118*x^4 + x^5.
%t MathematicaP[1, x_] := 8 + 9*x + x^2;
P[2, x_] := 22 + 63*x + 24*x^2 + x^3;
P[3, x_] := 52 + 373*x + 314*x^2 + 55*x^3 + x^4;
P[4, x_] := 114 + 1739*x + 3252*x^2 + 1265*x^3 + 118*x^4 + x^5;
P[n_, x_] := P[n, x] = Expand[Together[((12*n^3 + 4*n^2 - 16*n + (6*n^3 - 10*n^2 - 20*n)*x - (14*n^2 + 14*n)*x^2 - (3*n^3 - 2*n^2 - 9*n)*x^3)*P[n - 4, x] + ((6*n^3 + 2*n^2 - 8*n)*x - (6*n^2 + 6*n)*x^2 - (3*n^3 - 2*n^2 - 9*n)*x^3)*D[P[n - 4, x], x] + (12*n^3 - 14*n^2 - 22*n + 24 + (6*n^3 - 16*n^2 - 16*n + 6)*x - (25*n^2 - 6*n - 35)*x^2 - (3*n^3 - 5*n^2 + 9*n + 25)*x^3)*P[n - 3, x] + (18*n^2 + 6*n - 24 + (6*n^3 - 22*n^2 - 34*n + 14)*x - (15*n^2 - 18*n - 41)*x^2 - (3*n^3 - 8*n^2 - 11*n + 12)*x^3 + (3*n^2 - 2*n - 9)*x^4)*D[P[n - 3, x], x] + ((6*n^2 + 2*n - 8)*x - (6*n^2 + 8*n - 2)*x^2 - (3*n^2 - 8*n - 15)*x^3 + (3*n^2 - 2*n - 9)*x^4)*D[P[n - 3, x], {x, 2}] - (6*n^2 + 14*n + 12 + (12*n^2 - 36*n - 84)*x + (5*n^2 - 10*n + 25)*x^2 + 16*n*x^3)*P[n - 2, x] - (6*n^2 + 2*n - 8 - 20*x - (3*n^2 + 4*n - 35)*x^2 + 4*x^3 - 16*x^4)*D[P[n - 2, x], x] + (8*x^2 - 16*x^3 + 8*x^4)*D[P[n - 2, x], {x, 2}] + (6*n + 26 + (16*n - 28)*x + 16*x^2)*P[n - 1, x])/(8)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 8 + 9*x + x^2; v[2] = 22 + 63*x + 24*x^2 + x^3; v[3] = 52 + 373*x + 314*x^2 + 55*x^3 + x^4; v[4] = 114 + 1739*x + 3252*x^2 + 1265*x^3 + 118*x^4 + x^5; for(n = 5, N, my(i = n); v[i] = ((12*n^3 + 4*n^2 - 16*n + (6*n^3 - 10*n^2 - 20*n)*x - (14*n^2 + 14*n)*x^2 - (3*n^3 - 2*n^2 - 9*n)*x^3)*v[i - 4] + ((6*n^3 + 2*n^2 - 8*n)*x - (6*n^2 + 6*n)*x^2 - (3*n^3 - 2*n^2 - 9*n)*x^3)*deriv(v[i - 4], x) + (12*n^3 - 14*n^2 - 22*n + 24 + (6*n^3 - 16*n^2 - 16*n + 6)*x - (25*n^2 - 6*n - 35)*x^2 - (3*n^3 - 5*n^2 + 9*n + 25)*x^3)*v[i - 3] + (18*n^2 + 6*n - 24 + (6*n^3 - 22*n^2 - 34*n + 14)*x - (15*n^2 - 18*n - 41)*x^2 - (3*n^3 - 8*n^2 - 11*n + 12)*x^3 + (3*n^2 - 2*n - 9)*x^4)*deriv(v[i - 3], x) + ((6*n^2 + 2*n - 8)*x - (6*n^2 + 8*n - 2)*x^2 - (3*n^2 - 8*n - 15)*x^3 + (3*n^2 - 2*n - 9)*x^4)*deriv(deriv(v[i - 3], x), x) - (6*n^2 + 14*n + 12 + (12*n^2 - 36*n - 84)*x + (5*n^2 - 10*n + 25)*x^2 + 16*n*x^3)*v[i - 2] - (6*n^2 + 2*n - 8 - 20*x - (3*n^2 + 4*n - 35)*x^2 + 4*x^3 - 16*x^4)*deriv(v[i - 2], x) + (8*x^2 - 16*x^3 + 8*x^4)*deriv(deriv(v[i - 2], x), x) + (6*n + 26 + (16*n - 28)*x + 16*x^2)*v[i - 1])/(8)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.?
References
- [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.