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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 ${\displaystyle {\text{gcd}}\left(n^{2},{\frac {n!}{\left\lfloor n/2\right\rfloor !^{2}}}\right)}$ 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 ${\displaystyle {\text{lcm}}\left(n^{2},{\frac {n!}{\left\lfloor n/2\right\rfloor !^{2}}}\right)}$ 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 ${\displaystyle {\frac {{\text{gcd}}\left(n^{2},n!\right)}{{\text{gcd}}\left(n^{2},{\frac {n!}{\left\lfloor n/2\right\rfloor !^{2}}}\right)}}}$ 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 ${\displaystyle {\frac {{\text{lcm}}\left(n^{2},n!\right)}{{\text{lcm}}\left(n^{2},{\frac {n!}{\left\lfloor n/2\right\rfloor !^{2}}}\right)}}\quad (n>0;\ a(0)=1)}$ 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 ${\displaystyle {\text{lcm}}\left(n^{2},n!\right)}$ 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 ${\displaystyle {\text{Denominator}}[(-2)^{n}B_{n}]}$ 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 ${\displaystyle \sigma _{n}(x)}$ (formula (6.52) in Concrete Mathematics) at x = 1.  ${\displaystyle \sigma _{n}(1)=\sum _{k=1}^{2^{n-1}}T(n,k)^{-1}.}$  Let ${\displaystyle B_{n}}$ denote the Bernoulli number, then ${\displaystyle B_{n}=n!\sigma _{n}(1)}$ 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