A129891
Sum of coefficients of polynomials defined in comments lines.
Original entry on oeis.org
1, 2, 4, 9, 20, 44, 96, 209, 455, 991, 2159, 4704, 10249, 22330, 48651, 105997, 230938, 503150, 1096225, 2388372, 5203604, 11337218, 24700671, 53815949, 117250109, 255455647, 556567394, 1212606837, 2641935832, 5756049469, 12540844137
Offset: 0
- Paul Curtz, Gazette des Mathématiciens, 1992, no. 52, p. 44.
Sums of coefficients of polynomials defined in
A140530.
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R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^3)/(1-3*x+2*x^2-x^4) )); // G. C. Greubel, Oct 24 2023
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a:= n-> (Matrix([1, 1, 0, 1]). Matrix(4, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0, 1][i] else 0 fi)^n)[1, 1]:
seq(a(n), n=0..50); # Alois P. Heinz, Oct 14 2009
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u[n_ /; n < 3] = 1; u[n_] := 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] // Total, {n, 0, 30}] (* Jean-François Alcover, Oct 02 2012 *)
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def A129891_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( (1-x+x^3)/(1-3*x+2*x^2-x^4) ).list()
A129891_list(40) # G. C. Greubel, Oct 24 2023
A141411
Defined in comments.
Original entry on oeis.org
3, 1, 31, 28, 365, 514, 4388, 8220, 53871, 122284, 673222, 1748055, 8535397, 24383499, 109449848, 334783855, 1415768769, 4548229589, 18434398665, 61345927764, 241210652738, 823296868656, 3167642169823, 11010462627756, 41708741708554, 146886286090602
Offset: 0
- P. Curtz, Gazette des Mathematiciens, 1992, 52, p.44.
- P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, 55, pp.67-78 .
-
u:= proc(n) Digits:= max(n+10);
trunc (10* frac(evalf(Pi*10^(n-1))))
end:
P:= proc(n) option remember; local i, x;
if n=0 then u(0)
else unapply(expand(u(n)+x*add(u(i)*P(n-i-1)(x), i=0..n-1)), x)
fi
end:
a:= n-> (P(n)(1)+P(n)(-1))/2:
seq(a(n), n=0..30); # Alois P. Heinz, Sep 06 2009
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nmax = 25; digits = RealDigits[Pi, 10, nmax+1][[1]]; p[0][] = digits[[1]]; p[n][x_] := p[n][x] = digits[[n+1]] + x*Sum[digits[[i+1]] p[n-i-1][x], {i, 0, n-1}]; a[n_] := (p[n][1] + p[n][-1])/2; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Nov 22 2012 *)
A100597
Based on the first matrix inverse of transformed Bernoulli numbers as defined in the Comments line.
Original entry on oeis.org
1, 1, 2, 5, 14, 49, 258, 1385, 1342, -13739, 1727362, 20549165, -892047378, -13084315271, 979519187138, 16158974238545, -1747908612654946, -32246548780758179, 4903305033480792642, 100032668564662494485, -20685044415403212103730, -462550882810484735564351
Offset: 1
a(3) = 2, because M = [1; -1/2 1; 1/6 -1 1; ...], M^(-1) = [1; 1/2 1; 1/3 1 1; ...], and (1/3)*3! = 2.
- P. Curtz, Gazette des Mathematiciens, 1992, 52, p.44.
- P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, 55, pp.67-78.
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P:= proc(n) option remember; local i, u, x; u:= bernoulli; `if`(n=0, u(0), unapply(expand(u(n) +x *add(u(i) *P(n-i-1)(x), i=0..n-1)), x)) end: a:= n-> (1/Matrix(n, (i, j)-> coeff(P(i-1)(x), x, j-1)))[n, 1] *n!: seq(a(n), n=1..30); # Alois P. Heinz, Oct 12 2009
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p[0, x_] = BernoulliB[0]; p[n_, x_] := p[n, x] = BernoulliB[n] + x*Sum[BernoulliB[i]*p[n-i-1, x], {i, 0, n-1}]; t[m_] := Table[ PadRight[CoefficientList[p[n, x], x], m+1], {n, 0, m}]; mmax = 20; Inverse[t[mmax-1]][[All, 1]]*Range[mmax]!
(* Jean-François Alcover, Jun 29 2011 *)
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Comments