A141904 -id:A141904 - cp's OEIS frontend

A142048 Triangle of the denominators of the coefficients [x^k] P(n,x) defined in A141904.

1, 3, 1, 5, 3, 1, 7, 45, 1, 1, 9, 105, 15, 3, 1, 11, 1575, 945, 15, 3, 1, 13, 10395, 175, 945, 9, 1, 1, 15, 315315, 17325, 14175, 189, 15, 3, 1, 17, 45045, 23648625, 7425, 2835, 189, 15, 3, 1, 19, 6891885, 1576575, 23648625, 93555, 945, 135, 45, 1, 1, 21, 72747675, 7309575

Offset: 0

Paul Curtz, Sep 14 2008

Are all the denominators odd?

Cf. A141904 (numerators).

ClearAll[u, p]; u[n_] := (-1)^n/(2*n + 1); p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[u[i]*p[n - i - 1][x] , {i, 0, n-1}] // Expand; row[n_] := CoefficientList[ p[n][x], x]; Table[row[n], {n, 0, 10}] // Flatten // Denominator (* Jean-François Alcover, Oct 02 2012 *)

Edited and extended by R. J. Mathar, Aug 24 2009

A152650 Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of certain polynomials P(n,x) given below.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 9, 4, 1, 1, 2, 9, 8, 5, 1, 1, 4, 27, 32, 25, 6, 1, 1, 4, 81, 32, 125, 18, 7, 1, 1, 8, 81, 128, 625, 36, 49, 8, 1, 1, 2, 243, 256, 625, 54, 343, 32, 9, 1, 1, 4, 729, 1024, 3125, 324, 2401, 256, 81, 10, 1

Offset: 0

Paul Curtz, Dec 10 2008

Let the polynomials P be defined by P(0,x)=u(0), P(n,x)= u(n) + x*Sum_{i=0..n-1} u(i)*P(n-i-1,x) and coefficients u(i)=1/i!. These u are reminiscent of the Taylor expansion of exp(x). Then P(n,x) = Sum_{k=0..n} c(n,k)*x^k.
n!*P(n,x) are the row polynomials of A152818. - Peter Bala, Oct 09 2011
Conjecture: All roots of P(n,x) are real, hence negative. - Jean-François Alcover, Oct 10 2012

			The triangle c(n,k) and polynomials start in row n = 0 as:
1 = 1;
1, 1 = 1 + x;
1/2, 2, 1 = 1/2 + 2*x + x^2;
1/6, 2, 3, 1, = 1/6+2*x+3*x^2+x^3
1/24, 4/3, 9/2, 4, 1, = 1/24 + 4/3*x + 9/2*x^2 + 4*x^3 + x^4;
1/120, 2/3, 9/2, 8, 5, 1, = 1/120 + 2/3*x + 9/2*x^2 + 8*x^3 + 5*x^4 + x^5;
1/720, 4/15, 27/8, 32/3, 25/2, 6, 1, = 1/720 + 4/15*x + 27/8*x^2 + 32/3*x^3 + 25/2*x^4 + 6*x^5 + x^6;
1/5040, 4/45, 81/40, 32/3, 125/6, 18, 7, 1 = 1/5040 + 4/45*x + 81/40*x^2 + 32/3*x^3 + 125/6*x^4 + 18*x^5 + 7*x^6 + x^7;

Jean-François Alcover, Plot showing roots of P(20,x)

Cf. A152656 (denominators), A140749, A141412, A141904, A142048. A152818.

u := proc(i) 1/i! end:
P := proc(n,x) option remember ; if n =0 then u(0); else u(n)+x*add( u(i)*procname(n-1-i,x),i=0..n-1) ; expand(%) ; fi; end:
A152650 := proc(n,k) p := P(n,x) ; numer(coeftayl(p,x=0,k)) ; end:
seq(seq(A152650(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Aug 24 2009

ClearAll[u, p]; u[n_] := 1/n!; p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[u[i]*p[n-i-1][x] , {i, 0, n-1}] // Expand; row[n_] := CoefficientList[p[n][x], x]; Table[row[n], {n, 0, 10}] // Flatten // Numerator (* Jean-François Alcover, Oct 02 2012 *)

Edited and extended by R. J. Mathar, Aug 24 2009

A231121 Denominators of coefficients of expansion of arctan(x)^3.

Original entry on oeis.org

1, 1, 15, 945, 175, 17325, 23648625, 1576575, 7309575, 1283268987, 3360942585, 1932541986375, 135664447443525, 218461268025, 242856109621125, 27604644460267875, 4479480941961650625, 1151866527932995875, 31580724596338947904875, 809762169136896100125, 4742892704944677157875

Offset: 0

Jean-François Alcover and Paul Curtz, Nov 04 2013

Cf. A002428, A002429, A071968, A140749, A141904, A142048, A081051.

a[n_] := SeriesCoefficient[ArcTan[x]^3, {x, 0, 2*n+3}] // Denominator
(* or *) a[n_] := 3*Sum[2^(i-2)*Binomial[2*(n+1), i-1]*StirlingS1[i, 3]/i!, {i, 3, 2n+3}] // Denominator; Table[a[n], {n, 0, 20}] (* from the formula given by Ruperto Corso in A002429 *)
Take[Denominator[CoefficientList[Series[ArcTan[x]^3,{x,0,50}],x] ], {4,-1,2}] (* Harvey P. Dale, Apr 07 2017 *)

Let u(n) = (-1)^n/(2*n+1) and P(n,x) = u(n) + x*Sum_{i=0..n-1} u(i)*P(n-i-1,x), with P(0,x) = u(0). Then, the terms are the denominators of the coefficients of x^2 in each polynomial.

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A142048 Triangle of the denominators of the coefficients [x^k] P(n,x) defined in A141904.

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A152650 Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of certain polynomials P(n,x) given below.

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A231121 Denominators of coefficients of expansion of arctan(x)^3.

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