A192951 -id:A192951 - cp's OEIS frontend

A193047 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 2, 19, 102, 377, 1104, 2777, 6282, 13155, 25998, 49153, 89792, 159681, 278034, 476131, 804790, 1346457, 2234768, 3686201, 6051290, 9897491, 16143262, 26275009, 42698112, 69304897, 112393634, 182155507, 295080582, 477850745

Offset: 0

Clark Kimberling, Jul 15 2011

nonn

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n^4, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Index entries for linear recurrences with constant coefficients, signature (6,-14,15,-5,-4,4,-1).

Cf. A192232, A192744, A192951, A193046.

Mathematica
```
(See A193046.)
```

a(n)=6*a(n-1)-14*a(n-2)+15*a(n-3)-5*a(n-4)-4*a(n-5)+4*a(n-6)-a(n-7).

G.f.: -x*(-1+4*x-21*x^2-x^3-6*x^4+x^5) / ( (x^2+x-1)*(x-1)^5 ). - R. J. Mathar, May 12 2014

A193049 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 1, 2, 4, 12, 37, 105, 268, 625, 1355, 2772, 5414, 10188, 18605, 33161, 57954, 99683, 169265, 284452, 474066, 784852, 1292567, 2119923, 3465620, 5651323, 9197673, 14947276, 24263704, 39353486, 63787101, 103341963, 167366400, 270986619

Offset: 0

Clark Kimberling, Jul 15 2011

nonn

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n(4-5*n^2+n^4)/120, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Index entries for linear recurrences with constant coefficients, signature (7,-20,29,-20,1,8,-5,1).

Cf. A192232, A192744, A192951, A193048.

Mathematica
```
(See A193048.)
```

a(n)=7*a(n-1)-20*a(n-2)+29*a(n-3)-20*a(n-4)+*a(n-5)+8*a(n-6)-5*a(n-7)+a(n-8).

G.f.: -x*(x^2-x+1)*(x^4-5*x^3+9*x^2-5*x+1) / ( (x^2+x-1)*(x-1)^6 ). - R. J. Mathar, May 12 2014

A193041 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 3, 13, 44, 122, 292, 631, 1267, 2411, 4408, 7820, 13560, 23109, 38867, 64721, 106964, 175782, 287660, 469275, 763795, 1241071, 2014128, 3265848, 5292144, 8571817, 13879587, 22468981, 36368252, 58859186, 95251828, 154138015

Offset: 0

Clark Kimberling, Jul 15 2011

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+1+n^3, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

Cf. A192232, A192744, A192951, A193008.

Mathematica
```
(See A193008.)
```

a(n) = 5*a(n-1)-9*a(n-2)+6*a(n-3)+a(n-4)-3*a(n-5)+a(n-6).

G.f.: -x*(7*x^2-2*x+1)/((x-1)^4*(x^2+x-1)). [Colin Barker, Nov 12 2012]

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A193047 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

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A193049 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A193041 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula