A192232 -id:A192232 - cp's OEIS frontend

A192239 Coefficient of x in the reduction of the polynomial x(x+1)(x+2)...(x+n-1) by x^2 -> x+1.

0, 1, 3, 13, 71, 463, 3497, 29975, 287265, 3042545, 35284315, 444617525, 6048575335, 88347242335, 1378930649745, 22903345844335, 403342641729665, 7506843094993825, 147226845692229875, 3034786640911840925, 65592491119118514375

Offset: 1

Clark Kimberling, Jun 26 2011

nonn

See A192232.

Cf. A192232, A192238.

q[x_] := x + 1;
p[0, x_] := 1; p[1, x_] := x;
p[n_, x_] := (x + n) p[n - 1, x] /; n > 1
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[
  Last[Most[
    FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,
   20}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}]  (* A192238 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}]  (* A192239 *)
(* by Peter J. C. Moses, Jun 25 2011 *)
Flatten[{0,RecurrenceTable[{a[n]==2*(n-1)*a[n-1]-(n^2-3*n+1)*a[n-2],a[2]==1,a[3]==3},a,{n,2,20}]}] (* or *)
Flatten[{0,FullSimplify[Rest[Rest[CoefficientList[Series[-1/10*(Sqrt[5]+3)*Sqrt[5]*(x-1)^(Sqrt[5]/2-1/2)/(-1)^((1/2)*Sqrt[5]-1/2)-(1/10)*(Sqrt[5]-3)*Sqrt[5]*(x-1)^(-Sqrt[5]/2-1/2)/(-1)^(-Sqrt[5]/2-1/2), {x, 0, 20}], x]* Range[0, 20]!]]]}] (* Vaclav Kotesovec, Oct 20 2012 *)

From Vaclav Kotesovec, Oct 20 2012: (Start)
Recurrence (for n>3): a(n) = 2*(n-1)*a(n-1) - (n^2-3*n+1)*a(n-2).
E.g.f.: (for n>1): -1/10*(sqrt(5) + 3)*sqrt(5)*(x-1)^(sqrt(5)/2 - 1/2)/(-1)^((1/2)*sqrt(5) - 1/2) - (1/10)*(sqrt(5) - 3)*sqrt(5)*(x-1)^(-sqrt(5)/2 - 1/2)/(-1)^(-sqrt(5)/2 - 1/2).
a(n) ~ n!*n^(sqrt(5)/2 - 1/2)*(3*sqrt(5) - 5)/(10*Gamma((1 + sqrt(5))/2)).
(End)

Minor edits by Vaclav Kotesovec, Mar 31 2014

A192380 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

0, 2, 4, 20, 60, 230, 776, 2792, 9720, 34410, 120780, 425788, 1497716, 5274190, 18562320, 65348560, 230024944, 809742418, 2850375060, 10033806180, 35320352940, 124333050422, 437670231064, 1540664252600, 5423363437800, 19091038878650, 67203259647836

Offset: 1

Clark Kimberling, Jun 29 2011

The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+2). For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=2+x+3x^2 -> 5+4x
p(3,x)=8x+4x^2+4x^3 -> 8+20x
p(4,x)=4+4x+21x^2+10x^3+5x^4 -> 45+60x.
From these, read
A192379=(1,0,5,8,45,...) and A192380=(0,2,4,20,60,...)

Index entries for linear recurrences with constant coefficients, signature (2,6,-2,-1).

Cf. A192232, A192379, A192381.

Mathematica
```
(See A192379.)
```

a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: 2*x^2 / (x^4+2*x^3-6*x^2-2*x+1). [Colin Barker, Dec 09 2012]

A192464 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) = 1 + x^n + x^(2n).

Original entry on oeis.org

2, 4, 7, 16, 38, 95, 242, 624, 1619, 4216, 11002, 28747, 75170, 196652, 514607, 1346880, 3525566, 9229063, 24160402, 63250168, 165586907, 433505384, 1134920882, 2971243731, 7778788418, 20365086100, 53316412567, 139584058864, 365435613974, 956722540271

Offset: 1

Clark Kimberling, Jul 01 2011

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232. The coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) = 1 + x^n + x^(2n) is 2*A051450.

			The first four polynomials p(n,x) and their reductions are as follows:
p(1,x) = 1 + x   + x^2 ->  2 +  2x
p(2,x) = 1 + x^2 + x^4 ->  4 +  4x
p(3,x) = 1 + x^3 + x^6 ->  7 + 10x
p(4,x) = 1 + x^4 + x^8 -> 16 + 24x.
From these, read
A192464 = (2, 4, 7, 16, ...) and 2*A051450 = (2, 4, 10, 24, ...).

Index entries for linear recurrences with constant coefficients, signature (5,-7,1,3,-1).

Cf. A000045, A192232, A051450.

Remove["Global`*"];
q[x_] := x + 1; p[n_, x_] := 1 + x^n + x^(2 n);
Table[Simplify[p[n, x]], {n, 1, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192464 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* 2*A051450 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]
(* A051450 *)
Table[1-Fibonacci[n]+Fibonacci[1+n]-Fibonacci[2n]+Fibonacci[1+2n], {n, 1, 29}]
(* Friedjof Tellkamp, Nov 22 2021 *)

G.f.: -x*(3*x^4-7*x^3-x^2+6*x-2)/((x-1)*(x^2-3*x+1)*(x^2+x-1)). - Colin Barker, Nov 12 2012

a(n) = 1 - Fibonacci(n) + Fibonacci(1+n) - Fibonacci(2n) + Fibonacci(1+2n). - Friedjof Tellkamp, Nov 22 2021

A192465 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x)=1+x^n+x^(2n).

Original entry on oeis.org

3, 9, 25, 93, 353, 1389, 5505, 21933, 87553, 349869, 1398785, 5593773, 22372353, 89483949, 357924865, 1431677613, 5726666753, 22906579629, 91626143745, 366504225453, 1466016202753, 5864063412909, 23456250855425, 93824997829293

Offset: 1

Clark Kimberling, Jul 01 2011

nonn

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			The first four polynomials p(n,x) and their reductions are as follows:
p(1,x)=1+x+x^2 -> 3+2x
p(2,x)=1+x^2+x^4 -> 9+6x
p(3,x)=1+x^3+x^6 -> 25+24x
p(4,x)=1+x^4+x^8 -> 93+90x.
From these, read
A192465=(3,9,25,93,...) and A192466=(2,6,24,90,...)

Cf. A192232, A192466, A192464.

Remove["Global`*"];
q[x_] := x + 2; p[n_, x_] := 1 + x^n + x^(2 n);
Table[Simplify[p[n, x]], {n, 1, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192465 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192466 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]
(* A192467 *)

Empirical G.f.: -x*(3*x-1)*(8*x^2-3)/((x-1)*(x+1)*(2*x-1)*(4*x-1)). [Colin Barker, Nov 12 2012]

A192761 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

0, 1, 5, 11, 22, 40, 70, 119, 199, 329, 540, 882, 1436, 2333, 3785, 6135, 9938, 16092, 26050, 42163, 68235, 110421, 178680, 289126, 467832, 756985, 1224845, 1981859, 3206734, 5188624, 8395390, 13584047, 21979471, 35563553, 57543060, 93106650

Offset: 0

Clark Kimberling, Jul 09 2011

The titular polynomial is defined recursively by p(n,x) = x*(n-1,x) + n + 3 for n > 0, where p(0,x) = 1. For discussions of polynomial reduction, see A192232 and A192744.

Construct a triangle with T(n,0) = n*(n+1)+1 and T(n,n) = (n+1)*(n+2)/2 starting at n=0. Define the interior terms by T(r,c) = T(r-2,c-1) + T(r-1,c). The sequence of its row sums is 1, 6, 17, 39, 79, 149, 268, 467,... and the first differences of these (the sum of the terms in row(n) less those in row(n-1)) equals a(n+1). - J. M. Bergot, Mar 10 2013

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A192744, A192232, partial sums of A022318.

q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 3;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
  (* A022318 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192761 *)

a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). G.f.: x*(2*x^2-2*x-1) / ((x-1)^2*(x^2+x-1)). [Colin Barker, Dec 08 2012]

A192808 Constant term in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2. See Comments.

Original entry on oeis.org

1, 2, 6, 26, 126, 618, 3022, 14746, 71902, 350538, 1708910, 8331130, 40615294, 198004778, 965298958, 4705957722, 22942154782, 111845982474, 545263681710, 2658231220538, 12959222223038, 63177890368490, 308000415667278, 1501542003033370

Offset: 0

Clark Kimberling, Jul 10 2011

For discussions of polynomial reduction, see A192232 and A192744.
If the reduction (x^2 + c)^n by x^3 -> x^2 + c is applied to the polynomials (x^2+c)^n for c=1 instead of c=2, the results are as follows:
A052554: constant terms,
A052529: coefficients of x,
A124820: coefficients of x^2.
Those three sequences satisfy the recurrence:
u(n) = 4*u(n-1) - 3*u(n-2) + u(n-3).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-12,8).

Cf. A192744, A192232, A192808 - A192811.

a:=[1,2,6];; for n in [4..25] do a[n]:=7*a[n-1]-12*a[n-2]+8*a[n-3]; od; Print(a); # Muniru A Asiru, Jan 02 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3) )); // G. C. Greubel, Jan 02 2019

q = x^3; s = x^2 + 2; z = 40;
p[n_, x_] := (x^2 + 2)^n;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192808 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192809 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}] (* A192810 *)
uu = u2/2  (* A192811 *)
LinearRecurrence[{7,-12,8}, {1,2,6}, 50] (* G. C. Greubel, Jan 02 2019 *)

my(x='x+O('x^30)); Vec((1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3)) \\ G. C. Greubel, Jan 02 2019

((1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 02 2019

a(n) = 7*a(n-1) - 12*a(n-2) + 8*a(n-3).

G.f.: (1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3). - Colin Barker, Jul 26 2012

A192960 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

1, 3, 7, 15, 29, 53, 93, 159, 267, 443, 729, 1193, 1945, 3163, 5135, 8327, 13493, 21853, 35381, 57271, 92691, 150003, 242737, 392785, 635569, 1028403, 1664023, 2692479, 4356557, 7049093, 11405709, 18454863, 29860635, 48315563, 78176265

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A192232, A192744, A192951, A192961.

F:=Fibonacci;; List([0..40], n-> 2*F(n+4)-(2*n+5)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [2*F(n+4)-(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n - 1, x] + n^2 + 2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192960 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192961 *)
(* Second program *)
With[{F=Fibonacci}, Table[2*F[n+4]-(2*n+5), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

vector(40, n, n--; f=fibonacci; 2*f(n+4)-(2*n+5)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [2*f(n+4)-(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From R. J. Mathar, May 09 2014: (Start)
G.f.: (1+x)*(1-x+x^2)/((1-x-x^2)*(1-x)^2).
a(n) - a(n-1) = A019274(n+2). (End)
a(n) = 2*Fibonacci(n+4) - (2*n + 5). - G. C. Greubel, Jul 12 2019

A192967 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

1, 0, 2, 4, 9, 17, 31, 54, 92, 154, 255, 419, 685, 1116, 1814, 2944, 4773, 7733, 12523, 20274, 32816, 53110, 85947, 139079, 225049, 364152, 589226, 953404, 1542657, 2496089, 4038775, 6534894, 10573700, 17108626, 27682359, 44791019, 72473413, 117264468

Offset: 0

Clark Kimberling, Jul 13 2011

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A192232, A192744, A192951.

List([0..40], n-> 3*Fibonacci(n+1) -n-2); # G. C. Greubel, Jul 11 2019

I:=[1, 0, 2, 4]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012

[3*Fibonacci(n+1) -n-2: n in [0..40]]; // G. C. Greubel, Jul 11 2019

q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n*(n-1)/2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192967 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192968 *)
LinearRecurrence[{3,-2,-1,1}, {1,0,2,4}, 41] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2012 *)
Table[3*Fibonacci[n+1] -n-2, {n,0,40}] (* G. C. Greubel, Jul 11 2019 *)

vector(40, n, n--; f=fibonacci; 3*f(n+1)-n-2) \\ G. C. Greubel, Jul 11 2019

[3*fibonacci(n+1) -n-2 for n in (0..40)] # G. C. Greubel, Jul 11 2019

a(0)=1, a(1)=0, for n > 1, a(n) = a(n-1) + a(n-2) + n - 1. - Alex Ratushnyak, May 10 2012
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 -3*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014
a(n) = 3*Fibonacci(n+1) - n - 2. - G. C. Greubel, Jul 11 2019

A192971 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

1, 2, 9, 21, 44, 83, 149, 258, 437, 729, 1204, 1975, 3225, 5250, 8529, 13837, 22428, 36331, 58829, 95234, 154141, 249457, 403684, 653231, 1057009, 1710338, 2767449, 4477893, 7245452, 11723459, 18969029, 30692610, 49661765, 80354505

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000032, A000045, A192232, A192744, A192951, A192972.

F:=Fibonacci;; List([0..40], n-> 5*F(n+3)+F(n+1) -2*(2*n+5)); # G. C. Greubel, Jul 24 2019

F:=Fibonacci; [5*F(n+3)+F(n+1) -2*(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 24 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + 2*n^2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192971 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192972 *)
(* Additional programs *)
With[{F = Fibonacci}, Table[5*F[n+3]+F[n+1] -2*(2*n+5), {n,0,40}]] (* G. C. Greubel, Jul 24 2019 *)

vector(40, n, n--; f=fibonacci; 5*f(n+3)+f(n+1) -2*(2*n+5)) \\ G. C. Greubel, Jul 24 2019

f=fibonacci; [5*f(n+3)+f(n+1) -2*(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 24 2019

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1-x+5*x^2-x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014
a(n) = 4*Fibonacci(n+3) + Lucas(n+2) - 2*(2*n+5). - G. C. Greubel, Jul 24 2019

A087124 a(n) = Fibonacci(n) + Fibonacci(2n+1).

Original entry on oeis.org

1, 3, 6, 15, 37, 94, 241, 623, 1618, 4215, 11001, 28746, 75169, 196651, 514606, 1346879, 3525565, 9229062, 24160401, 63250167, 165586906, 433505383, 1134920881, 2971243730, 7778788417, 20365086099, 53316412566, 139584058863

Offset: 0

Paul Barry, Aug 15 2003

Binomial transform of A087123.

For n>=1, a(n) is the coefficient of x in the reduction by x^2->x+1 of the polynomial 1+x^n+x^(2n+1). For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232. - Clark Kimberling, Jul 01 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Cf. A000045, A087123, A192232.

[Fibonacci(n)+Fibonacci(2*n+1): n in [0..40]]; // Vincenzo Librandi, Mar 13 2012

CoefficientList[Series[(1-2*x)*(1+x-x^2)/((1-3*x+x^2)*(1-x-x^2)),{x,0,1001}],x] (* Vincenzo Librandi, Mar 13 2012 *)
LinearRecurrence[{4,-3,-2,1},{1,3,6,15},30] (* Harvey P. Dale, Aug 17 2024 *)

a(n)=fibonacci(n)+fibonacci(2*n+1) \\ Charles R Greathouse IV, Mar 13 2012

G.f.: (1-2*x)*(1+x-x^2)/((1-3*x+x^2)*(1-x-x^2)). - Colin Barker, Mar 12 2012

cp's OEIS Frontend

A192239 Coefficient of x in the reduction of the polynomial x(x+1)(x+2)...(x+n-1) by x^2 -> x+1.

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A192380 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

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A192464 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) = 1 + x^n + x^(2n).

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A192465 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x)=1+x^n+x^(2n).

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A192761 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

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A192808 Constant term in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2. See Comments.

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A192960 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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