A192744 -id:A192744 - cp's OEIS frontend

A192815 Coefficient of x in the reduction of the polynomial (2*x + 1)^n by x^3 -> x^2 + x + 1.

0, 2, 4, 14, 72, 346, 1612, 7526, 35216, 164786, 770964, 3606974, 16875480, 78953226, 369388508, 1728211222, 8085563168, 37828901730, 176985297700, 828038725486, 3874040046440, 18124981139642, 84799056637292, 396738620092614

Offset: 0

Clark Kimberling, Jul 10 2011

For discussions of polynomial reduction, see A192232 and A192744.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (5,-3,7).

Cf. A192744, A192232, A192815, A192816.

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( 2*x*(1-3*x)/(1-5*x+3*x^2-7*x^3) )); // G. C. Greubel, Jan 03 2019

(See A192814.)
LinearRecurrence[{5,-3,7}, {0,2,4}, 30] (* G. C. Greubel, Jan 03 2019 *)

concat([0], Vec(2*x*(1-3*x)/(1-5*x+3*x^2-7*x^3)+O(x^30))) \\ Charles R Greathouse IV, Jul 11 2011

(2*x*(1-3*x)/(1-5*x+3*x^2-7*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 03 2019

a(n) = 5*a(n-1) - 3*a(n-2) + 7*a(n-3).
a(n) = 2*A192816(n).
G.f.: 2*x*(1-3*x)/(1-5*x+3*x^2-7*x^3). - Bruno Berselli, Jul 11 2011

A192816 a(n) = A192815(n)/2.

Original entry on oeis.org

0, 1, 2, 7, 36, 173, 806, 3763, 17608, 82393, 385482, 1803487, 8437740, 39476613, 184694254, 864105611, 4042781584, 18914450865, 88492648850, 414019362743, 1937020023220, 9062490569821, 42399528318646, 198369310046307, 928085399264344

Offset: 0

Clark Kimberling, Jul 10 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (5,-3,7).

Cf. A192242, A192744, A192814, A192815.

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1-3*x)/(1-5*x+3*x^2-7*x^3) )); // G. C. Greubel, Jan 03 2019

(* See A192814. *)
LinearRecurrence[{5,-3,7}, {0,1,2}, 30] (* G. C. Greubel, Jan 03 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-3*x)/(1-5*x+3*x^2-7*x^3))) \\ G. C. Greubel, Jan 03 2019

(x*(1-3*x)/(1-5*x+3*x^2-7*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 03 2019

a(n) = 5*a(n-1) - 3*a(n-2) + 7*a(n-3).

G.f.: x*(1-3*x)/(1-5*x+3*x^2-7*x^3). - Bruno Berselli, Jul 11 2011

A192873 Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) given in Comments.

Original entry on oeis.org

0, 1, 2, 7, 18, 49, 128, 337, 882, 2311, 6050, 15841, 41472, 108577, 284258, 744199, 1948338, 5100817, 13354112, 34961521, 91530450, 239629831, 627359042, 1642447297, 4299982848, 11257501249, 29472520898, 77160061447, 202007663442, 528862928881, 1384581123200

Offset: 0

Clark Kimberling, Jul 11 2011

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = x*p(n-1,x) + (x^2)*p(n-1,x) + 1. See A192872.

First differences give A236428. - Richard R. Forberg, Feb 23 2014

			The coefficients of all the polynomials p(n,x) are Fibonacci numbers (A000045).  The first 6 and their reductions:
p(0,x) = 1 -> 1
p(1,x) = x -> x
p(2,x) = 1 +2*x^2 -> 3 +2*x
p(3,x) = 1 +x +3*x^3 -> 4 +7*x
p(4,x) = 1 +x +2*x^2 +5*x^4 -> 13 +18*x
p(5,x) = 1 +x +2*x^2 +3*x^3 +8*x^5 -> 30 +49*x
G.f. = x + 2*x^2 + 7*x^3 + 18*x^4 + 49*x^5 + 128*x^6 + 337*x^7 + ...

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

Cf. A192872, A192232, A192744.

a:=[0,1,2,7];; for n in [5..40] do a[n]:=3*a[n-1]-3*a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 07 2019

I:=[0,1,2,7]; [n le 4 select I[n] else 3*Self(n-1) - 3*Self(n-3) +Self(n-4): n in [1..40]]; // G. C. Greubel, Jan 07 2019

seq(coeff(series(x*(x^2-x+1)/((1-x)*(1+x)*(x^2-3*x+1)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Jan 08 2019

(See A192872.)
a[ n_] := SeriesCoefficient[ x * (1 - x + x^2) / ((1 - x^2) * (1 - 3*x + x^2)), {x, 0, Abs @ n}]; (* Michael Somos, Apr 08 2014 *)
LinearRecurrence[{3,0,-3,1}, {0,1,2,7}, 40] (* G. C. Greubel, Jan 07 2019 *)

concat(0, Vec(-x*(x^2-x+1)/((x-1)*(x+1)*(x^2-3*x+1)) + O(x^40))) \\ Colin Barker, Apr 01 2014

(x*(x^2-x+1)/((1-x^2)*(x^2-3*x+1))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jan 07 2019

a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
G.f.: x*(x^2-x+1) / ((1-x)*(1+x)*(x^2-3*x+1)). - Colin Barker, Apr 01 2014
a(n) = (1/10) * (4L(2*n) - 3*(-1)^n - 5), with L(n) the Lucas numbers (A000032). - Ralf Stephan, Apr 06 2014
a(-n) = a(n) for all n in Z. - Michael Somos, Apr 08 2014

More terms from Colin Barker, Apr 01 2014

A192905 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

0, 1, 3, 8, 25, 79, 248, 777, 2435, 7632, 23921, 74975, 234992, 736529, 2308483, 7235416, 22677769, 71078319, 222778856, 698249753, 2188505347, 6859373216, 21499148257, 67384199871, 211200478176, 661959956001, 2074763216131

Offset: 0

Clark Kimberling, Jul 12 2011

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x. For details, see A192904.

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

Cf. A192232, A192744, A192904, A192872.

a:=[0,1,3,8];; for n in [5..30] do a[n]:=3*a[n-1]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 11 2019

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1-x^2)/(1-3*x-x^3-x^4) )); // G. C. Greubel, Jan 11 2019

(See A192904.)
LinearRecurrence[{3,0,1,1}, {0,1,3,8}, 30] (* G. C. Greubel, Jan 11 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-x^2)/(1-3*x-x^3-x^4))) \\ G. C. Greubel, Jan 11 2019

(x*(1-x^2)/(1-3*x-x^3-x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 11 2019

a(n) = 3*a(n-1) + a(n-3) + a(n-4).

G.f.: x*(1-x)*(1+x)/(1-3*x-x^3-x^4). - Colin Barker, Aug 31 2012

A192908 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 1, 3, 7, 17, 43, 111, 289, 755, 1975, 5169, 13531, 35423, 92737, 242787, 635623, 1664081, 4356619, 11405775, 29860705, 78176339, 204668311, 535828593, 1402817467, 3672623807, 9615053953, 25172538051, 65902560199

Offset: 0

Clark Kimberling, Jul 12 2011

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x + 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A192232, A192744, A192872, A192904.
Cf. A000045; A052995: 2*Fibonacci(2*n-1) for n>0.
CF. A055588, A097136.

Concatenation([1], List([1..30], n -> 1+2*Fibonacci(2*(n-1)))); # G. C. Greubel, Jan 11 2019

[1] cat [1+2*Fibonacci(2*(n-1)): n in [1..30]]; // G. C. Greubel, Jan 11 2019

u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 0; f = 1;
q = x^2; s = u*x + v; z = 26;
p[0, x_] := a;  p[1, x_] := b*x + c
p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
Table[Expand[p[n, x]], {n, 0, 8}]
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}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]    (* A192908 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]    (* A069403 *)
Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *)
Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *)
LinearRecurrence[{4,-4,1}, {1,1,3,7}, 30] (* G. C. Greubel, Jan 11 2019 *)

vector(30, n, n--; if(n==0,1,1+2*fibonacci(2*n-2))) \\ G. C. Greubel, Jan 11 2019

[1]+[1+2*fibonacci(2*(n-1)) for n in (1..30)] # G. C. Greubel, Jan 11 2019

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n>3.
G.f.: 1 + x*(1 - x - x^2)/((1 - x)*(1 - 3*x + x^2)). - R. J. Mathar, Jul 13 2011
a(n) = 2*Fibonacci(2*n-2) + 1 for n>0, a(0)=1. - Bruno Berselli, Dec 27 2016
a(n) = -1 + 3*a(n-1) - a(n-2) with a(1) = 1 and a(2) = 3. Cf. A055588 and A097136. - Peter Bala, Nov 12 2017

A192909 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 1, 3, 9, 27, 83, 259, 811, 2541, 7963, 24957, 78221, 245165, 768413, 2408415, 7548629, 23659463, 74155215, 232422687, 728476151, 2283243129, 7156307287, 22429820697, 70301181369, 220343094521, 690615411545, 2164577236699

Offset: 0

Clark Kimberling, Jul 12 2011

nonn

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x) + 1, with p(0,x) = 1, p(1,x) = x + 1.

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

Cf. A192232, A192744, A192872, A192904, A192910.

a:=[1,1,3,9,27];; for n in [6..30] do a[n]:=4*a[n-1]-3*a[n-2] +a[n-3]-a[n-5]; od; a; # G. C. Greubel, Jan 11 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (x^2-x+1)*(x^2+2*x-1)/((1-x)*(x^4+x^3+3*x-1)) )); // G. C. Greubel, Jan 11 2019

u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 1; f = 1;
q = x^2; s = u*x + v; z = 24;
p[0, x_] := a;  p[1, x_] := b*x + c
p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
Table[Expand[p[n, x]], {n, 0, 8}]
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}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192909 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192910 *)
Simplify[FindLinearRecurrence[u0]]
Simplify[FindLinearRecurrence[u1]]
LinearRecurrence[{4,-3,1,0,-1}, {1,1,3,9,27}, 30] (* G. C. Greubel, Jan 11 2019 *)

my(x='x+O('x^30)); Vec((x^2-x+1)*(x^2+2*x-1)/((1-x)*(x^4+x^3+3*x -1))) \\ G. C. Greubel, Jan 11 2019

((x^2-x+1)*(x^2+2*x-1)/((1-x)*(x^4+x^3+3*x-1))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 11 2019

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

G.f.: (x^2-x+1)*(x^2+2*x-1) / ( (1-x)*(x^4+x^3+3*x-1) ). - R. J. Mathar, Jul 13 2011

A192910 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

0, 1, 4, 13, 42, 133, 418, 1311, 4110, 12883, 40380, 126563, 396684, 1243317, 3896896, 12213937, 38281814, 119985657, 376067806, 1178699171, 3694364986, 11579148423, 36292212248, 113749700903, 356522616120, 1117439209033, 3502359540252

Offset: 0

Clark Kimberling, Jul 12 2011

nonn

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x) + 1, with p(0,x) = 1, p(1,x) = x + 1.

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

Cf. A192232, A192744, A192872, A192904, A192909.

a:=[0,1,4,13,42];; for n in [6..30] do a[n]:=4*a[n-1]-3*a[n-2] + a[n-3]-a[n-5]; od; a; # G. C. Greubel, Jan 12 2019

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1+x)*(1-x+x^2)/((1-x)*(1-3*x-x^3-x^4)) )); // G. C. Greubel, Jan 12 2019

(See A192909.)
LinearRecurrence[{4,-3,1,0,-1}, {0,1,4,13,42}, 30] (* G. C. Greubel, Jan 12 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1+x)*(1-x+x^2)/((1-x)*(1-3*x -x^3-x^4)))) \\ G. C. Greubel, Jan 12 2019

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

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

G.f.: x*(1+x)*(1-x+x^2)/((1-x)*(1-3*x-x^3-x^4)). - R. J. Mathar, Jul 13 2011

A192911 Constant term in the reduction by (x^3 -> x + 1) of the polynomial F(n+1)*x^n, where F(n)=A000045 (Fibonacci sequence).

Original entry on oeis.org

1, 0, 0, 3, 5, 16, 52, 147, 442, 1320, 3916, 11664, 34717, 103298, 307440, 914949, 2722885, 8103424, 24116008, 71769885, 213589298, 635647790, 1891705884, 5629770720, 16754357925, 49861446392, 148389084968, 441610143507

Offset: 0

Clark Kimberling, Jul 12 2011

Regarding polynomial reduction, see A192232 and A192744.  In the case of the reduction at A192911, each term in the three resulting sequences is a product of a Fibonacci number and a tribonacci numbers
A192911(n) = F(n+1)*T3(n+1), where F=A000045, T3=A000073.
A192912(n) = F(n+1)*T2(n), where T2=A001590.
A192913(n) = F(n+1)*T3(n).
All three obey the same linear recurrence, shown below at Formula.

			The first six polynomials and reductions:
1 -> 1
x -> 2
2*x^2 -> 2*x^2
3*x^3 -> 3 + 3*x + 3*x^2
5*x^4 -> 5 + 10*x + 10*x^2
8*x^5 -> 16 + 24*x + 32*x^2

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,5,2,-1,1).

Cf. A192232, A192744, A192912, A192913.

a:=[1,0,0,3,5,16];; for n in [7..30] do a[n]:=a[n-1]+4*a[n-2] + 5*a[n-3]+2*a[n-4]-a[n-5]+a[n-6]; od; a; # G. C. Greubel, Jan 12 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (x+1)*(2*x^2+2*x-1)/(x^6-x^5+2*x^4+5*x^3+4*x^2+x-1) )); // G. C. Greubel, Jan 12 2019

q = x^3; s = x^2 + x + 1; z = 22;
p[0, x_] := 1; p[1, x_] := x;
p[n_, x_] := p[n - 1, x]*x + p[n - 2, x]*x^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}] (* A192911 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192912 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}] (* A192913 *)
LinearRecurrence[{1,4,5,2,-1,1},{1,0,0,3,5,16},28] (* Ray Chandler, Aug 02 2015 *)

my(x='x+O('x^30)); Vec((x+1)*(2*x^2+2*x-1)/(x^6-x^5+2*x^4+5*x^3 +4*x^2+x-1)) \\ G. C. Greubel, Jan 12 2019

((x+1)*(2*x^2+2*x-1)/(x^6-x^5+2*x^4+5*x^3+4*x^2+x-1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019

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

G.f.: (x+1)*(2*x^2+2*x-1)/(x^6-x^5+2*x^4+5*x^3+4*x^2+x-1). - Colin Barker, Aug 31 2012

A192941 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x)=(2x+1)(2x+2)...(2x+n).

Original entry on oeis.org

1, 1, 6, 38, 276, 2276, 21032, 215336, 2419824, 29611120, 391950240, 5579965600, 85018056640, 1380373170880, 23792373137280, 433881469662080, 8346202841391360, 168894762064666880, 3586667489988830720, 79753496814542988800

Offset: 0

Clark Kimberling, Jul 13 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(0,x) = 1
p(1,x) = 2x+1 -> 1+2x
p(2,x) = (2x+1)(2x+2) -> 6+10x
p(3,x) = (2x+1)(2x+2)(2x+3) -> 38+62x
From these, read
A192941=(1,2,6,38,...) and A192942=(0,2,10,62,...)

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A192232, A192744, A192942.

SetDefaultRealField(RealField(100)); R:= RealField(); s:=Sqrt(5); [Round(s*(s-1)*Gamma(n+2+s)/Gamma(s+2) + Sin(Pi(R)*(s+3))*Gamma(s+2) *Gamma(n+2-s)/(Pi(R)*(s-1)))/10: n in [0..20]]; // G. C. Greubel, Jul 25 2019

(* First program *)
q = x^2; s = x + 1; z = 26;
p[0, x]:= 1;
p[n_, x_]:= (2*x + n)*p[n-1, x];
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}] (* A192941 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192942 *)
u2/2 (* A192950 *)
(* Additional programs *)
With[{s = Sqrt[5]}, Table[FullSimplify[(s*(s-1)*Gamma[n+2+s]/Gamma[s+2] + Sin[Pi*(s+3)]*Gamma[s+2]*Gamma[n+2-s]/(Pi*(s-1)))/10], {n, 0, 20}]] (* G. C. Greubel, Jul 25 2019 *)

default(realprecision, 100); vector(20, n, n--; s=sqrt(5); round((s^2-s)*gamma(n+2+s)/gamma(s+2) + sin(Pi*(s+3))*gamma(s+2)* gamma(n+2-s)/(Pi*(s-1)))/10 ) \\ G. C. Greubel, Jul 25 2019

s=sqrt(5); [round(s*(s-1)*gamma(n+2+s)/gamma(s+2) + sin(pi*(s+3))* gamma(s+2)*gamma(n+2-s)/(pi*(s-1)))/10 for n in (0..20)] # G. C. Greubel, Jul 25 2019

a(n) = 1/10*(5-sqrt(5))*Gamma(n+2+sqrt(5))/Gamma(2+sqrt(5)) + 1/10*sin(Pi*(3+sqrt(5)))*(1+sqrt(5))*Gamma(1+sqrt(5))*Gamma(n+2-sqrt(5))/(Pi*(sqrt(5)-1)). - Vaclav Kotesovec, Oct 26 2012

Conjecture: a(n) +(-2*n-1)*a(n-1) +(n^2-5)*a(n-2)=0. - R. J. Mathar, May 08 2014

A192942 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x)=(2x+1)(2x+2)...(2x+n).

Original entry on oeis.org

0, 2, 10, 62, 448, 3688, 34056, 348568, 3916352, 47919520, 634256480, 9029234720, 137569217280, 2233574372480, 38497936301440, 702049663399040, 13504656880506880, 273280886412413440, 5803407252377602560, 129044887279907315200

Offset: 0

Clark Kimberling, Jul 13 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(0,x) = 1
p(1,x) = 2x+1 -> 1+2x
p(2,x) = (2x+1)(2x+2) -> 6+10x
p(3,x) = (2x+1)(2x+2)(2x+3) -> 38+62x
From these, read
A192941=(1,2,6,38,...) and A192942=(0,2,10,62,...)

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A192232, A192744, A192941, A192950.

SetDefaultRealField(RealField(100)); R:= RealField(); s:=Sqrt(5); [Round(s*Gamma(n+2+s)/Gamma(s+2) - Sin(Pi(R)*(s+3))*Gamma(s+1) *Gamma(n+2-s)/(Pi(R)*(s-1)))/5: n in [0..20]]; // G. C. Greubel, Jul 25 2019

(* First program *)
q = x^2; s = x + 1; z = 26;
p[0, x]:= 1;
p[n_, x_]:= (2x+n)*p[n-1, x];
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}] (* A192941 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192942 *)
u2/2 (* A192950 *)
(* Second program *)
With[{s = Sqrt[5]}, Table[FullSimplify[(s*Gamma[n+2+s]/Gamma[s+2] - Sin[Pi*(s+3)]*Gamma[s+1]*Gamma[n+2-s]/(Pi*(s-1)))/5], {n, 0, 20}]] (* G. C. Greubel, Jul 26 2019 *)

default(realprecision, 100); vector(20, n, n--; s=sqrt(5); round(s*gamma(n+2+s)/gamma(s+2) - sin(Pi*(s+3))*gamma(s+1)*gamma(n+2-s)/(Pi*(s-1)))/5 ) \\ G. C. Greubel, Jul 25 2019

s=sqrt(5); [round(s*gamma(n+2+s)/gamma(s+2) - sin(pi*(s+3))* gamma(s+1)*gamma(n+2-s)/(pi*(s-1)))/5 for n in (0..20)] # G. C. Greubel, Jul 25 2019

Conjecture: a(n) +(-2*n-1)*a(n-1) +(n^2-5)*a(n-2)=0. - R. J. Mathar, May 08 2014

cp's OEIS Frontend

A192815 Coefficient of x in the reduction of the polynomial (2*x + 1)^n by x^3 -> x^2 + x + 1.

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A192816 a(n) = A192815(n)/2.

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A192873 Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) given in Comments.

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A192905 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

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A192908 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

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A192909 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

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