A277000
Numerators of an asymptotic series for the Gamma function (even power series).
Original entry on oeis.org
1, -1, 19, -2561, 874831, -319094777, 47095708213409, -751163826506551, 281559662236405100437, -49061598325832137241324057, 5012066724315488368700829665081, -26602063280041700132088988446735433, 40762630349420684160007591156102493590477
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!:
A277000 := n -> numer(b(2*n)): seq(A277000(n), n=0..12);
# Alternatively the rational sequence by recurrence:
R := proc(n) option remember; local k; `if`(n=0, 1,
add(bernoulli(2*m+2,1/2)* R(n-m-1)/(2*m+1), m=0..n-1)/(2*n)) end:
seq(numer(R(n)), n=0..12); # Peter Luschny, Sep 30 2016
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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_] := Numerator[b[2n]];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Sep 09 2018 *)
A181855
Numerator of Nemes numbers G_n.
Original entry on oeis.org
1, 1, 1, 239, -46409, 9113897, -695818219549, 5649766313929, -1070083202835456443, 93856597276403726428217, -4815785492460413153189484781, 674781102986061046417681986493, -9845646538265462155478818981872958283
Offset: 0
G_0 = 1, G_1 = 1/12, G_2 = 1/1440, G_3 = 239/362880.
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G := proc(n) option remember; local k; `if`(n=0,1,
add(bernoulli(2*m+2)*G(n-m-1)/(2*m+1),m=0..n-1)/(2*n)) end;
a181855 := n -> numer(G(n));
# Alternatively:
p := n -> CompleteBellB(n, 0, seq((k-2)!*bernoulli(k, 1), k=2..n))/n!:
a := n -> numer(p(2*n)): seq(a(n), n=0..12); # Peter Luschny, Oct 03 2016
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a[0] = 1; a[n_] := a[n] = Sum[ BernoulliB[2m + 2]*a[n - m - 1]/(2m + 1), {m, 0, n}]/(2n); Table[a[n] // Numerator, {n, 0, 12}] (* Jean-François Alcover, Jul 26 2013 *)
CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
p[n_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, 1], {k, 2, n}]]]/n!;
a[n_] := Numerator[p[2n]];
Table[a[n], {n, 0, 12}] (* 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 1-3 of 3 results.
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