A166345: Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+2*x+x^2)/(1-x)^4, read by rows.
A166345 ↗ palindromicγ-positivenonnegative coefficients
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)interlacing
Recurrence from A166345
\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 + n - 2\,n x + n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)
\(P_{2}(x) = 1 + 2\,x + x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 1 | 2 | |||||||||||||
| 2 | 1 | 2 | 1 | 4 | ||||||||||||
| 3 | 1 | 7 | 7 | 1 | 16 | |||||||||||
| 4 | 1 | 18 | 42 | 18 | 1 | 80 | ||||||||||
| 5 | 1 | 41 | 198 | 198 | 41 | 1 | 480 | |||||||||
| 6 | 1 | 88 | 799 | 1584 | 799 | 88 | 1 | 3360 | ||||||||
| 7 | 1 | 183 | 2925 | 10331 | 10331 | 2925 | 183 | 1 | 26880 | |||||||
| 8 | 1 | 374 | 10056 | 58874 | 103310 | 58874 | 10056 | 374 | 1 | 241920 | ||||||
| 9 | 1 | 757 | 33160 | 305888 | 869794 | 869794 | 305888 | 33160 | 757 | 1 | 2419200 | |||||
| 10 | 1 | 1524 | 106293 | 1488832 | 6490186 | 10437528 | 6490186 | 1488832 | 106293 | 1524 | 1 | 26611200 | ||||
| 11 | 1 | 3059 | 334119 | 6911965 | 44361586 | 108056470 | 108056470 | 44361586 | 6911965 | 334119 | 3059 | 1 | 319334400 | |||
| 12 | 1 | 6130 | 1036006 | 30989050 | 284015615 | 1003231508 | 1512790580 | 1003231508 | 284015615 | 30989050 | 1036006 | 6130 | 1 | 4151347200 | ||
| 13 | 1 | 12273 | 3181578 | 135352266 | 1729968575 | 8575529583 | 18615386124 | 18615386124 | 8575529583 | 1729968575 | 135352266 | 3181578 | 12273 | 1 | 58118860800 | |
| 14 | 1 | 24560 | 9704283 | 579588000 | 10138717801 | 68752863248 | 207487469115 | 297846177984 | 207487469115 | 68752863248 | 10138717801 | 579588000 | 9704283 | 24560 | 1 | 871782912000 |
b-files: A166345, A166345 (reversed triangle) ?
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Technical details
- Node id
n-5c39e26cb9304567· revision 0 · JSON- Definition source
oeis_catalog:A166345- Fingerprint
f6-c9655e7f89d7b43e
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: yes
- γ-positive (gamma_positive): yes
- Nonnegative coefficients (
nonn): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): no
Matches ?
- A166345 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
- A166345 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 2, 1; 3: 1, 7, 7, 1; 4: 1, 18, 42, 18, 1; 5: 1, 41, 198, 198, 41, 1; 6: 1, 88, 799, 1584, 799, 88, 1; 7: 1, 183, 2925, 10331, 10331, 2925, 183, 1; 8: 1, 374, 10056, 58874, 103310, 58874, 10056, 374, 1; 9: 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1;
%F FormulaThe row polynomials satisfy (n-2)*P(n,x) = (n-2+(n^2-2*n)*x)*P(n-1,x) + ((n-2)*x-(n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 2*x + x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n^2 - 2*n)*x)*P[n - 1, x] + ((n - 2)*x - (n - 2)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 2*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n^2 - 2*n)*x)*v[i - 1] + ((n - 2)*x - (n - 2)*x^2)*deriv(v[i - 1], x))/(n - 2)); 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
A166345 triangle, offset 1, from P0
%e ExampleTriangle begins: 1: 1; 2: 1, 1; 3: 1, 2, 1; 4: 1, 7, 7, 1; 5: 1, 18, 42, 18, 1; 6: 1, 41, 198, 198, 41, 1; 7: 1, 88, 799, 1584, 799, 88, 1; 8: 1, 183, 2925, 10331, 10331, 2925, 183, 1; 9: 1, 374, 10056, 58874, 103310, 58874, 10056, 374, 1; 10: 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1;
%F FormulaThe row polynomials satisfy (n-3)*P(n,x) = (n-3+(n^2-4*n+3)*x)*P(n-1,x) + ((n-3)*x-(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 2*x + x^2.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n^2 - 4*n + 3)*x)*P[n - 1, x] + ((n - 3)*x - (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 2*x + x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n^2 - 4*n + 3)*x)*v[i - 1] + ((n - 3)*x - (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
A166345 reversed triangle, offset 1, from P0
%e ExampleTriangle begins: 1: 1; 2: 1, 1; 3: 1, 2, 1; 4: 1, 7, 7, 1; 5: 1, 18, 42, 18, 1; 6: 1, 41, 198, 198, 41, 1; 7: 1, 88, 799, 1584, 799, 88, 1; 8: 1, 183, 2925, 10331, 10331, 2925, 183, 1; 9: 1, 374, 10056, 58874, 103310, 58874, 10056, 374, 1; 10: 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1;
%F FormulaRow n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (n-3+(n^2-4*n+3)*x)*P(n-1,x) + ((n-3)*x-(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 2*x + x^2.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n^2 - 4*n + 3)*x)*P[n - 1, x] + ((n - 3)*x - (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 2*x + x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n^2 - 4*n + 3)*x)*v[i - 1] + ((n - 3)*x - (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); 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 ≥ 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 Liu–Wang theorem, Liu–Wang theorem with a lag row
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem, Liu–Wang theorem with a lag row
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 3
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m ≥ 4
Proof chains ? JSON
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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