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Stockpiling

A181861
GCD(squares, swinging factorials)
1, 1, 2, 3, 2, 5, 4, 7, 2, 9, 4, 11, 12, 13, 4, 45, 2, 17, 4, 19
Definition  \text{gcd}\left(n^2, \frac{n!}{\left\lfloor n /2 \right\rfloor!^2} \right)
Comment A181861(n) = gcd(A000290(n), A056040(n)).
Offset 0
Maple
 A181861 := n -> igcd(n^2,n!/iquo(n,2)!^2);
Crossrefs. A170826 A181860 A056040 A000290
List

1, 1, 2, 3, 2, 5, 4, 7, 2, 9, 4, 11, 12, 13, 4, 45, 2, 17, 4, 19, 4, 21, 4, 23, 4, 25, 4, 27, 8, 29, 180, 31, 2, 99, 4, 175, 12, 37, 4, 117, 20, 41, 12, 43, 8, 675, 4, 47, 36, 49, 4, 153, 8, 53, 4, 55, 56, 57, 4, 59, 16

A181860
LCM(squares, swinging factorials)
0, 1, 4, 18, 48, 150, 180, 980, 2240, 5670, 6300, 30492
Definition  \text{lcm}\left(n^2, \frac{n!}{\left\lfloor n /2 \right\rfloor!^2} \right)
Comment A181860(n) = lcm(A000290(n), A056040(n)).
Offset 0
Maple
 A181860 := n -> ilcm(n^2,n!/iquo(n,2)!^2);
Crossrefs. A170825 A181861 A056040 A000290
List

0, 1, 4, 18, 48, 150, 180, 980, 2240, 5670, 6300, 30492, 11088, 156156, 168168, 257400, 1647360, 3719430, 3938220, 17551820, 18475600, 81477396, 85357272, 373173528, 389398464

A181859
GCD(squares, factorials) /
GCD(squares, swinging factorials)
1, 1, 1, 1, 4, 1, 9, 1, 32, 9, 25, 1, 12, 1, 49, 5, 128, 1, 81
Definition  \frac {\text{gcd}\left(n^2, n! \right) }{ \text{gcd}\left(n^2, \frac{n!}{\left\lfloor n /2 \right\rfloor!^2} \right)}
Comment A181861(n) = A170826(n) / A181861(n).
Offset 0
Maple
 A181859 := n -> 
 igcd(n^2,n!)/igcd(n^2,n!/iquo(n,2)!^2);
Crossrefs. A170826 A181860 A056040 A000290
List

1, 1, 1, 1, 4, 1, 9, 1, 32, 9, 25, 1, 12, 1, 49, 5, 128, 1, 81, 1, 100, 21, 121, 1, 144, 25, 169, 27, 98, 1, 5, 1, 512, 11, 289, 7, 108, 1, 361, 13, 80, 1, 147, 1, 242, 3, 529, 1, 64, 49, 625, 17, 338, 1, 729, 55, 56, 57

A181858
LCM(squares, factorials) /
LCM(squares, swinging factorials)
1, 1, 1, 1, 1, 4, 4, 36, 18, 64, 576, 14400, 43200, 518400
Definition  \frac {\text{lcm}\left(n^2, n! \right) }{ \text{lcm}\left(n^2, \frac{n!}{\left\lfloor n /2 \right\rfloor!^2} \right)} \quad (n > 0; \  a(0) = 1)
Comment A181858(n) = A181857(n) / A181860(n).
Offset 0
Maple
 A181858 := n -> `if`(n=0, 1, 
 ilcm(n^2,n!)/ilcm(n^2,n!/iquo(n,2)!^2));
Crossrefs. A170826 A181860 A056040 A000290
List

1, 1, 1, 1, 1, 4, 4, 36, 18, 64, 576, 14400, 43200, 518400, 518400, 5080320, 12700800, 1625702400, 1625702400, 131681894400, 131681894400, 627056640000, 13168189440000, 1593350922240000

A181857
LCM(squares, factorials)
0, 1, 4, 18, 48, 600, 720, 35280, 40320, 362880
Definition  \text{lcm}\left( n^2, n! \right)
Comment A181857(n) = lcm(A000290(n), A000142(n)).
Offset 0
Maple
 A181857 := n -> ilcm(n^2, n!) ;
Crossrefs. A170826 A181860 A056040 A000290
List

0, 1, 4, 18, 48, 600, 720, 35280, 40320, 362880, 3628800, 439084800, 479001600, 80951270400, 87178291200, 1307674368000, 20922789888000, 6046686277632000

A markup style study

Note that the box below does not reproduce the official A141459. It reflects only what I think A141459 should say and what I think would be a better way to display the information visually.


A141459
Clausen Numbers / 2
1, 1, 3, 1, 15, 1, 21, 1, 15, 1, 33, 1, 1365
Definition Denominator[( − 2)nBn]
Comment a(2n+1) = 1; a(2n) = A001897(n).
Offset 0
Formula
Clausen := proc(n) local S, i;
S := numtheory[divisors](n); S := map(i->i+1, S);
S := select(isprime, S); mul(i, i=S) end; 
Generating function
Maple
 a := n -> denom((-2)^n * bernoulli(n)); 
 a := n -> `if`(n = 0, 1, Clausen(n) / 2);
References Thomas Clausen, Lehrsatz aus einer Abhandlung über die Bernoullischen Zahlen, Astr. Nachr. 17 (1840), 351-352.
Links Peter Luschny, Generalized Clausen numbers.
Crossrefs. A160014 A027760 A027642 A001897 A160035
Media list graph listen


If some boundary value of a formula f(n) (for example for n=0) does not give an integer value (for example (1/2)) it is much more sensible to replace the formula by ceil(f(n)) or floor(f(n)) then to decapitate the sequence.

See a response of Jaume Oliver i Lafont to this remark on the discussion page.

An alternative description of A106831

Note that the box below does not reproduce the official A106831. It reflects only what I think A106831 should say and what I think would be a better way to display the information visually.

A106831
S. C. Woon Numbers
1, 2, -6, 4, 24, -12, -12, 8, -120, 48, 36, -24, 48, -24,  -24, 16, 720, -240, -144, 96, -144, 72, 72, -48, -240, 96, 72, -48, 96, -48, -48, 32, -5040, 1440, 720, -480, 576
Definition Triangle read by rows. The numbers are generated by a tree-like algorithm as implemented below.
Comment T(n,k) are numbers used by S. C. Woon to compute the Stirling polynomials σn(x) (formula (6.52) in Concrete Mathematics) at x = 1.
  \sigma_n(1) = \sum_{k=1}^{2^{n-1}}  T(n, k)^{-1}. 

Let Bn denote the Bernoulli number, then Bn = nn(1) for n ≥ 0.

Example
Triangle begins: 
            1 
            2 
          -6,4 
       24,-12,-12,8 
-120,48,36,-24,48,-24,-24,16 
Maple
A106831_row := proc(n) local k,i,m,W,right,left,fact;
right:= proc(L) local i;
   [L[1],2, seq(L[i],i=2..nops(L))] end;
left := proc(L) local i;
   [-L[1],L[2]+1,seq(L[i],i=3..nops(L))] end;
fact := proc(L) local i;
   L[1]*mul(L[i]!,i=2..nops(L)) end;
W := array(0..2^n);
W[1] := [1,`if`(n=0,1,2)]; k := 2;
for i from 1 to n-1 do
   for m from k by 2 to 2*k-1 do
      W[m]   := left (W[iquo(m,2)]);
      W[m+1] := right(W[iquo(m,2)]);
    od;
    k := 2*k;
od;
seq(fact(W[i]),i=iquo(k,2)..k-1) end:
seq(print(A106831_row(i)),i=0..5);
References R. L. Graham, D. E. Knuth, O. Patashnik, 1989, Concrete Mathematics, Addison-Wesley.

S. C. Woon, A tree for generating Bernoulli numbers, Math. Mag., 70 (1997), 51-56.

Crossrefs.
Media list graph listen

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