A277001
Denominators of an asymptotic series for the Gamma function (even power series).
Original entry on oeis.org
1, 24, 5760, 2903040, 1393459200, 367873228800, 24103053950976000, 115694658964684800, 9440684171518279680000, 271211974879377138647040000, 3579998068407778230140928000000, 1976158933761093583037792256000000, 258955866680053703121272297226240000000
Offset: 0
The underlying rational sequence starts:
1, 0, -1/24, 0, 19/5760, 0, -2561/2903040, 0, 874831/1393459200, 0, ...
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b := n -> CompleteBellB(n,0,seq((k-2)!*bernoulli(k,1/2),k=2..n))/n!:
A277001 := n -> denom(b(2*n)): seq(A277001(n), n=0..12);
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CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
b[n_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, 1/2], {k, 2, n}]]]/n!;
a[n_] := Denominator[b[2n]];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Sep 09 2018 *)
A277002
Numerators of an asymptotic series for the Gamma function (odd power series).
Original entry on oeis.org
-1, 7, -31, 127, -511, 1414477, -8191, 118518239, -5749691557, 91546277357, -23273283019, 1982765468311237, -22076500342261, 455371239541065869, -925118910976041358111, 16555640865486520478399, -1302480594081611886641, 904185845619475242495834469891
Offset: 1
The underlying rational sequence b(n) starts:
0, -1/24, 0, 7/2880, 0, -31/40320, 0, 127/215040, 0, -511/608256, ...
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b := n -> `if`(n=0, 0, bernoulli(n+1, 1/2)/(n*(n+1))):
a := n -> numer(b(2*n-1)):
seq(a(n), n=1..18);
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b[n_] := BernoulliB[n+1, 1/2]/(n(n+1));
a[n_] := Numerator[b[2n-1]];
Array[a, 18] (* Jean-François Alcover, Sep 09 2018 *)
A277003
Denominators of an asymptotic series for the Gamma function (odd power series).
Original entry on oeis.org
24, 2880, 40320, 215040, 608256, 738017280, 1277952, 4010803200, 32006209536, 65745715200, 1736441856, 12641296711680, 10066329600, 12611097722880, 1337897345089536, 1086454927196160, 3401614098432, 83088011510887219200, 61022895341568
Offset: 1
The underlying rational sequence b(n) starts:
0, -1/24, 0, 7/2880, 0, -31/40320, 0, 127/215040, 0, -511/608256, ...
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b := n -> `if`(n=0, 0, bernoulli(n+1, 1/2)/(n*(n+1))):
a := n -> denom(b(2*n-1)):
seq(a(n), n=1..19);
-
b[n_] := BernoulliB[n+1, 1/2]/(n(n+1));
a[n_] := Denominator[b[2n-1]];
Array[a, 19] (* Jean-François Alcover, Sep 09 2018 *)
A276996
Numerators of coefficients of polynomials arising from applying the complete Bell polynomials to k!B_k(x)/(k*(k-1)) with B_k(x) the Bernoulli polynomials.
Original entry on oeis.org
1, 0, 0, 1, -1, 1, 0, 1, -3, 1, 1, -1, 6, -10, 5, 0, -1, -15, 95, -40, 16, 239, -1, 13, -85, 240, -237, 79, 0, 403, 21, 385, -1575, 3577, -2947, 421, -46409, -239, 3841, 175, 861, -8036, 45458, -10692, 2673, 0, -82451, -2657, 56177, 1638, 19488, -85260, 139656, -86472, 19216
Offset: 0
Polynomials start:
p_0(x) = 1;
p_1(x) = 0;
p_2(x) = 1/6 + -x + x^2;
p_3(x) = (1/2)*x + -(3/2)*x^2 + x^3;
p_4(x) = 1/60 + -x + 6*x^2 + -10*x^3 + 5*x^4;
p_5(x) = -(1/6)*x + -(15/2)*x^2 + (95/3)*x^3 + -40*x^4 + 16*x^5;
p_6(x) = 239/504 + -(1/4)*x + (13/4)*x^2 + -85*x^3 + 240*x^4 + -237*x^5 + 79*x^6;
Triangle starts:
1;
0, 0;
1, -1, 1;
0, 1, -3, 1;
1, -1, 6, -10, 5;
0, -1, -15, 95, -40, 16;
239,-1, 13, -85, 240, -237, 79;
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A276996_row := proc(n) local p;
p := (n,x) -> CompleteBellB(n,0,seq((k-2)!*bernoulli(k,x),k=2..n)):
seq(numer(coeff(p(n,x),x,k)), k=0..n) end:
seq(A276996_row(n), n=0..9);
# Recurrence for the polynomials:
A276996_poly := proc(n,x) option remember; local z;
if n = 0 then return 1 fi; z := proc(k) option remember;
if k=1 then 0 else (k-2)!*bernoulli(k,x) fi end;
expand(add(binomial(n-1,j)*z(n-j)*A276996_poly(j,x),j=0..n-1)) end:
for n from 0 to 5 do sort(A276996_poly(n,x)) od;
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CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
p[n_, x_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, x], {k, 2, n}]]];
row[0] = {1}; row[1] = {0, 0}; row[n_] := CoefficientList[p[n, x], x] // Numerator;
Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Sep 09 2018 *)
Showing 1-4 of 4 results.
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