A152656
Triangle read by rows: denominators of polynomials from A000142: P(0,x) = 1, P(n,x) = 1/n! + x*Sum_{i=0..n-1} P(n-i-1)/i!. Numerators are A152650.
Original entry on oeis.org
1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 24, 3, 2, 1, 1, 120, 3, 2, 1, 1, 1, 720, 15, 8, 3, 2, 1, 1, 5040, 45, 40, 3, 6, 1, 1, 1, 40320, 315, 80, 15, 24, 1, 2, 1, 1, 362880, 315, 560, 45, 24, 1, 6, 1, 1, 1, 3628800, 2835, 4480, 315, 144, 5, 24, 3, 2, 1, 1
Offset: 0
Contribution from _Vincenzo Librandi_, Dec 16 2012: (Start)
Triangle begins:
1,
1, 1,
2, 1, 1,
6, 1, 1, 1,
24, 3, 2, 1, 1,
120, 3, 2, 1, 1, 1,
720, 15, 8, 3, 2, 1, 1,
5040, 45, 40, 3, 6, 1, 1, 1,
40320, 315, 80, 15, 24, 1, 2, 1, 1,
362880, 315, 560, 45, 24, 1, 6, 1, 1, 1,
3628800, 2835, 4480, 315, 144, 5, 24, 3, 2, 1, 1,
...
First column: A000142; second column: A049606. (End)
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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 // Denominator (* Jean-François Alcover, Oct 02 2012 *)
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
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;
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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
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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 *)
A141904
Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of some polynomials P(n,x).
Original entry on oeis.org
1, -1, 1, 1, -2, 1, -1, 23, -1, 1, 1, -44, 14, -4, 1, -1, 563, -818, 22, -5, 1, 1, -3254, 141, -1436, 19, -2, 1, -1, 88069, -13063, 21757, -457, 43, -7, 1, 1, -11384, 16774564, -11368, 7474, -680, 56, -8, 1, -1, 1593269, -1057052, 35874836, -261502, 3982, -688, 212, -3, 1, 1, -15518938, 4651811
Offset: 0
The polynomials P(n,x) are for n=0 to 5:
1 = P(0,x).
-1/3+x = P(1,x).
1/5-2/3*x+x^2 = P(2,x).
-1/7+23/45*x-x^2+x^3 = P(3,x).
1/9-44/105*x+14/15*x^2-4/3*x^3+x^4 = P(4,x).
-1/11+563/1575*x-818/945*x^2+22/15*x^3-5/3*x^4+x^5 = P(5,x).
- P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p.44.
- P. Flajolet, X. Gourdon, B. Salvy, Gazette des Mathematiciens, 1993, no. 55, pp.67-78.
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u := proc(i) (-1)^i/(2*i+1) ; 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:
A141904 := proc(n,k) p := P(n,x) ; numer(coeftayl(p,x=0,k)) ; end: seq(seq(A141904(n,k),k=0..n),n=0..13) ; # R. J. Mathar, Aug 24 2009
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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 // Numerator (* Jean-François Alcover, Oct 02 2012 *)
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
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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 *)
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