| Displaying 1-5 of 5 results found.
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1
1, 1, 3, 2, 1, 3, 1, 4, 3, 1, 1, 6, 1, 1, 3, 4, 1, 3, 1, 2, 3, 1, 1, 12, 1, 1, 3, 2, 1, 3, 1, 4, 3, 1, 1, 6, 1, 1, 3, 4, 1, 3, 1, 2, 3, 1, 1, 12, 1, 1, 3, 2, 1, 3, 1, 4, 3, 1, 1, 6, 1, 1, 3, 4, 1, 3, 1, 2, 3, 1, 1, 12, 1, 1, 3, 2, 1, 3, 1, 4, 3, 1, 1, 6, 1, 1, 3, 4, 1, 3, 1, 2, 3, 1, 1, 12, 1, 1, 3, 2, 1, 3, 1, 4, 3
COMMENTS
Twice connected to Bernoulli numbers A164555/ A027642 via the Akiyama-Tanigawa algorithm. Conjecture (checked for the first 3000 entries): periodic with a(n+24)=a(n).
Is this a multiplicative function?
FORMULA
Multiplicative with a(2^3) = 2^min(e-1,2), a(3^e) = 3, and a(p^e) = 1 for a prime p >= 5.
Dirichlet g.f.: zeta(s) * (1 + 1/2^(2*s) + 1/2^(3*s-1)) * (1 + 2/3^s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/2. (End)
MAPLE
b := proc(n) n/(n+1)/(n+2) ; end: A051712 := proc(n) numer( b(n)-b(n+1)) ; end: A026741 := proc(n) if type(n, 'odd') then n; else n/2; fi; end:
Numerators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)*(a(n,k) - a(n,k+1)), n >= 0, k >= 0.
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22
1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 3, 1, -1, 1, 1, 2, 1, -1, 0, 1, 1, 5, 2, -3, -1, 1, 1, 1, 3, 5, -1, -1, 1, 0, 1, 1, 7, 5, 0, -4, 1, 1, -1, 1, 1, 4, 7, 1, -1, -1, 1, -1, 0, 1, 1, 9, 28, 49, -29, -5, 8, 1, -5, 5, 1, 1, 5, 3, 8, -7, -9, 5, 7, -5, 5, 0, 1, 1, 11, 15, 27, -28, -343, 295, 200, -44, -1017, 691, -691
FORMULA
a(n,k) = numerator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)/(j+k+1)).
E.g.f.: A(x,t) = (x+log(1-t))/(1-t-exp(-x)) = (1+(1/2)*x+(1/6)*x^2/2!-(1/30)*x^4/4!+...)*1 + (1/2+(1/3)*x+(1/6)*x^2/2!+...)*t + (1/3+(1/4)*x+(3/20)*x^2/2!+...)*t^2 + .... (End)
EXAMPLE
Table begins:
1 1/2 1/3 1/4 1/5 1/6 1/7 ...
1/2 1/3 1/4 1/5 1/6 1/7 ...
1/6 1/6 3/20 2/15 5/42 ...
0 1/30 1/20 2/35 5/84 ...
-1/30 -1/30 -3/140 -1/105 ...
Antidiagonals of numerator(a(n,k)):
1;
1, 1;
1, 1, 1;
1, 1, 1, 0;
1, 1, 3, 1, -1;
1, 1, 2, 1, -1, 0;
1, 1, 5, 2, -3, -1, 1;
1, 1, 3, 5, -1, -1, 1, 0;
1, 1, 7, 5, 0, -4, 1, 1, -1;
1, 1, 4, 7, 1, -1, -1, 1, -1, 0;
1, 1, 9, 28, 49, -29, -5, 8, 1, -5, 5;
MAPLE
a:= proc(n, k) option remember;
`if`(n=0, 1/(k+1), (k+1)*(a(n-1, k)-a(n-1, k+1)))
end:
seq(seq(numer(a(n, d-n)), n=0..d), d=0..12); # Alois P. Heinz, Apr 17 2013
MATHEMATICA
nmax = 12; a[0, k_]:= 1/(k+1); a[n_, k_]:= a[n, k]= (k+1)(a[n-1, k]-a[n-1, k+1]); Numerator[Flatten[Table[a[n-k, k], {n, 0, nmax}, {k, n, 0, -1}]]] (* Jean-François Alcover, Nov 28 2011 *)
PROG
(Magma)
function a(n, k)
if n eq 0 then return 1/(k+1);
else return (k+1)*(a(n-1, k) - a(n-1, k+1));
end if;
end function;
A051714:= func< n, k | Numerator(a(n, k)) >;
(SageMath)
def a(n, k):
if (n==0): return 1/(k+1)
else: return (k+1)*(a(n-1, k) - a(n-1, k+1))
def A051714(n, k): return numerator(a(n, k))
Denominators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)(a(n,k)-a(n,k+1)), n >= 0, k >= 0.
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19
1, 2, 2, 3, 3, 6, 4, 4, 6, 1, 5, 5, 20, 30, 30, 6, 6, 15, 20, 30, 1, 7, 7, 42, 35, 140, 42, 42, 8, 8, 28, 84, 105, 28, 42, 1, 9, 9, 72, 84, 1, 105, 140, 30, 30, 10, 10, 45, 120, 140, 28, 105, 20, 30, 1, 11, 11, 110, 495, 3960, 924, 231, 165, 220, 66, 66, 12, 12, 66, 55, 495, 264, 308, 132, 165, 44, 66, 1
FORMULA
a(n,k) = denominator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)/(j+k+1)). - Fabián Pereyra, Jan 14 2023
EXAMPLE
Table begins:
1 1/2 1/3 1/4 1/5 1/6 1/7 ...
1/2 1/3 1/4 1/5 1/6 1/7 ...
1/6 1/6 3/20 2/15 5/42 ...
0 1/30 1/20 2/35 5/84 ...
-1/30 -1/30 -3/140 -1/105 ...
MAPLE
a:= proc(n, k) option remember;
`if`(n=0, 1/(k+1), (k+1)*(a(n-1, k)-a(n-1, k+1)))
end:
seq(seq(denom(a(n, d-n)), n=0..d), d=0..12); # Alois P. Heinz, Apr 17 2013
MATHEMATICA
nmax = 12; a[0, k_] := 1/(k+1); a[n_, k_] := a[n, k] = (k+1)(a[n-1, k]-a[n-1, k+1]); Denominator[ Flatten[ Table[ a[n-k, k], {n, 0, nmax}, {k, n, 0, -1}]]](* Jean-François Alcover, Nov 28 2011 *)
1, 6, 6, 20, 15, 42, 28, 72, 45, 110, 66, 156, 91, 210, 120, 272, 153, 342, 190, 420, 231, 506, 276, 600, 325, 702, 378, 812, 435, 930, 496, 1056, 561, 1190, 630, 1332, 703, 1482, 780, 1640, 861, 1806, 946, 1980, 1035, 2162, 1128, 2352, 1225, 2550, 1326, 2756, 1431
COMMENTS
Also period length divided by 2 of pairs (a,b), where a has period 2*n-2 and b has period n.
0, 1/6, 1/6, 3/20, 2/15, 5/42, ...
-1/6, -1/60, 0, 1/420, 1/420, 1/504, ...
3/20, 1/60, 1/420, 0, -1/2520, -1/2520, ...
-2/15, -1/70, -1/420, -1/2520, 0, 1/13860, ...
5/42, 1/84, 1/504, 1/2520, -1/13860, 0, ...
Autosequence of the first kind. The main diagonal is A000004. The first two upper diagonals are equal. Their denominators are A000911. (End)
FORMULA
G.f.: (2*x^3+3*x^2+6*x+1)/(1-x^2)^3.
a(n) = (n+1)*(n+2) if n odd; or (n+1)*(n+2)/2 if n even = (n+1)*(n+2)*(3-(-1)^n)/4. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004
a(n) = lcm(2*n+2, n+2)/2.
E.g.f.: (2 + 8*x + x^2)*cosh(x)/2 + (2 + 2*x + x^2)*sinh(x). - Stefano Spezia, Apr 24 2024
MAPLE
seq((n+1)*(n+2)*(3-(-1)^n)/4, n=0..20); # C. Ronaldo
with(combinat): seq(lcm(n+1, binomial(n+2, n)), n=0..50); # Zerinvary Lajos, Apr 20 2008
MATHEMATICA
Table[LCM[2*n + 2, n + 2]/2, {n, 0, 40}] (* corrected by Amiram Eldar, Sep 14 2022 *) Denominator[#[[1]]/(#[[2]]#[[3]])&/@Partition[Range[0, 60], 3, 1]] (* Harvey P. Dale, Aug 15 2013 *)
PROG
(Haskell)
import Data.Ratio ((%), denominator)
a045896 n = denominator $ n % ((n + 1) * (n + 2))
Denominator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/ A045896.
+10
4
1, 60, 60, 70, 84, 504, 120, 990, 165, 572, 1092, 2730, 280, 4080, 2448, 1938, 855, 7980, 1540, 10626, 3036, 4600, 7800, 17550, 819, 21924, 12180, 8990, 7440, 32736, 5984, 39270, 5355, 15540, 25308, 54834, 4940, 63960, 34440
EXAMPLE
0, 1/60, 1/60, 1/70, 1/84, 5/504, 1/120, 7/990, 1/165, 3/572,...
MATHEMATICA
Denominator[#[[1]]-#[[2]]&/@(Partition[#[[1]]/(#[[2]]#[[3]])&/@Partition[ Range[50], 3, 1], 2, 1])] (* Harvey P. Dale, Nov 15 2014 *)
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