User:Peter Luschny/stock

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	${\text{Denominator}}[(-2)^{n}B_{n}]$
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 $\sigma _{n}(x)$ (formula (6.52) in Concrete Mathematics) at x = 1. $\sigma _{n}(1)=\sum _{k=1}^{2^{n-1}}T(n,k)^{-1}.$ Let $B_{n}$ denote the Bernoulli number, then $B_{n}=n!\sigma _{n}(1)$ 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