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User:Peter Luschny/stock
Stockpiling
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1, 1, 2, 3, 2, 5, 4, 7, 2, 9, 4, 11, 12, 13, 4, 45, 2, 17, 4, 19 | |
Definition | |
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 |
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0, 1, 4, 18, 48, 150, 180, 980, 2240, 5670, 6300, 30492 | |
Definition | |
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 |
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GCD(squares, swinging factorials) |
1, 1, 1, 1, 4, 1, 9, 1, 32, 9, 25, 1, 12, 1, 49, 5, 128, 1, 81 | |
Definition | |
Comment | A181861(n) = A170826(n) / A181861(n). |
Offset | 0 |
Maple |
A181859 := n -> |
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 |
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LCM(squares, swinging factorials) |
1, 1, 1, 1, 1, 4, 4, 36, 18, 64, 576, 14400, 43200, 518400 | |
Definition | |
Comment | A181858(n) = A181857(n) / A181860(n). |
Offset | 0 |
Maple |
A181858 := n -> `if`(n=0, 1, |
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 |
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0, 1, 4, 18, 48, 600, 720, 35280, 40320, 362880 | |
Definition | |
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 | |
1, 1, 3, 1, 15, 1, 21, 1, 15, 1, 33, 1, 1365 | |
Definition | |
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 | |
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 (formula (6.52) in Concrete Mathematics) at x = 1.
Let denote the Bernoulli number, then 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 |