Search: a050932 -id:a050932
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0, 0, 0, 1, 2, 5, 5, 7, 7, 5, 5, 11, 11, 91, 91, -9, -9, 1207, 1207, -10849, -10849, 65879, 65879, -783127, -783127, 61098739, 61098739, -2034290233, -2034290233, 72986324461, 72986324461
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OFFSET
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0,5
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COMMENTS
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Br(n)=0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, 0, -691/210, 0,.. .
a(n) is the numerators of Bp2(n)=0, 0, 0, 1, 2, 5/2, 5/2, 7/3, 7/3, 5/2, 5/2, 11/5, 11/5, 91/30, 91/30,... . Bp2(n) is an autosequence like Br(n).
With possible future sequences we can write the array PB
1, 0, 0, 0, 0, 0, 0, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0,
1, 3/2, 1, 0, 0, 0, 0, 0, 0,
1, 5/3, 2, 1, 0, 0, 0, 0, 0,
1, 5/3, 5/2, 5/2, 1, 0, 0, 0, 0,
1, 49/30, 5/2, 7/2, 3, 1, 0, 0, 0,
1, 49/30, 7/3, 7/2, 14/3, 7/2, 1, 0, 0,
1, 58/35, 7/3, 3, 14/3, 6, 4, 1, 0,
1, 58/35, 5/2, 3, 7/2, 6, 15/2, 9/2, 1, etc.
The first column is A000012. The second A165142(n+1)/(1 followed by A100650(n)). The third is Bp2(n+1). The next others are built by the same way. From the second,every column is based on A164555(n)/A027642(n).
With negative (2*n+2)-th diagonals,the array without 0's is the triangle NPB. The sum of every row is
1, 0, 1/2, -1/3, 1/3, -11/30, 11/30, -12/35, 12/35, -79/210, 79/210,... .
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LINKS
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EXAMPLE
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a(0)=a(1)=0, a(i)=numerators of 0+Br(0)=0, 0+Br(1)=1, 1+Br(2)=2, 2+Br(3)=5/2, 5/2+Br(4)=5/2,... .
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MATHEMATICA
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nmax = 30; Br[0] = 0; Br[1] = Br[2] = 1; Br[n_] := Numerator[2*n*BernoulliB[n-1]] / Denominator[n*BernoulliB[n-1]]; Bp2 = Join[{0, 0}, Table[Br[n], {n, 0, nmax-2}] // Accumulate]; a[n_] := Numerator[Bp2[[n+1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 18 2013 *)
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A002427
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Numerator of (2n+1) B_{2n}, where B_n are the Bernoulli numbers.
(Formerly M2510 N0993)
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+10
11
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1, 1, -1, 1, -3, 5, -691, 35, -3617, 43867, -1222277, 854513, -1181820455, 76977927, -23749461029, 8615841276005, -84802531453387, 90219075042845, -26315271553053477373, 38089920879940267, -261082718496449122051, 1520097643918070802691
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OFFSET
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0,5
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REFERENCES
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A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 73.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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(n+1)*B_n gives: 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...
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MAPLE
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gf := z / (1 - exp(-z)): ser := series(gf, z, 84):
seq(numer((n+1)!*coeff(ser, z, n)), n=0..42, 2); # Peter Luschny, Aug 29 2020
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MATHEMATICA
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Table[Numerator[2(2n+1)BernoulliB[2n]], {n, 1, 30}]
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PROG
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(PARI) a(n) = numerator((2*n+1)*bernfrac(2*n)); \\ Michel Marcus, Aug 06 2017
(Magma) [Numerator((2*n+1)*Bernoulli(2*n)): n in [1..30]]; // G. C. Greubel, Jul 03 2019
(Sage) [numerator((2*n+1)*bernoulli(2*n)) for n in (1..30)] # G. C. Greubel, Jul 03 2019
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CROSSREFS
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KEYWORD
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sign,easy,nice,frac
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AUTHOR
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STATUS
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approved
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A006955
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Denominator of (2n+1) B_{2n}, where B_n are the Bernoulli numbers.
(Formerly M1562)
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+10
11
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1, 2, 6, 6, 10, 6, 210, 2, 30, 42, 110, 6, 546, 2, 30, 462, 170, 6, 51870, 2, 330, 42, 46, 6, 6630, 22, 30, 798, 290, 6, 930930, 2, 102, 966, 10, 66, 1919190, 2, 30, 42, 76670, 6, 680862, 2, 690, 38874, 470, 6, 46410, 2, 330, 42, 106, 6, 1919190
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OFFSET
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0,2
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COMMENTS
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Also denominators of asymptotic expansion of polygamma function psi''(z).
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 260, (6.4.13).
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 73.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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Apparently a(n) = denominator(Sum_{k=0..2*n-1} (-1)^(2*n-k+1)*E1(2*n, k+1)/ binomial(2*n, k+1)), where E1(n, k) denotes the first-order Eulerian numbers A123125. - Peter Luschny, Feb 17 2021
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EXAMPLE
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(n+1)*B_n gives the sequence 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...
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MAPLE
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gf := z / (1 - exp(-z)): ser := series(gf, z, 220):
seq(denom((n+1)!*coeff(ser, z, n)), n=0..108, 2); # Peter Luschny, Aug 29 2020
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MATHEMATICA
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Denominator[Table[(2n+1)BernoulliB[2n], {n, 0, 60}]] (* Harvey P. Dale, Nov 03 2011 *)
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PROG
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(PARI) a(n) = denominator((2*n+1)*bernfrac(2*n)); \\ Michel Marcus, Aug 06 2017
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CROSSREFS
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KEYWORD
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nonn,frac,easy,nice
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AUTHOR
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STATUS
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approved
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A127187
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Nearest integer to (n+1)*Bernoulli(n).
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+10
11
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1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -3, 0, 18, 0, -121, 0, 1044, 0, -11112, 0, 142419, 0, -2164506, 0, 38488964, 0, -791648701, 0, 18649007091, 0, -498838420314, 0, 15036512507141, 0, -507331242588268, 0, 19044960439970134, 0
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OFFSET
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0,13
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LINKS
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A127188
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Nearest integer to (2*n+1)*Bernoulli(2*n).
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+10
11
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1, 1, 0, 0, 0, 1, -3, 18, -121, 1044, -11112, 142419, -2164506, 38488964, -791648701, 18649007091, -498838420314, 15036512507141, -507331242588268, 19044960439970134, -791159753019542794
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OFFSET
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0,7
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LINKS
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A050925
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Numerator of (n+1)*Bernoulli(n).
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+10
10
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1, -1, 1, 0, -1, 0, 1, 0, -3, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -1181820455, 0, 76977927, 0, -23749461029, 0, 8615841276005, 0, -84802531453387, 0, 90219075042845, 0, -26315271553053477373, 0, 38089920879940267, 0
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OFFSET
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0,9
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COMMENTS
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The denominators are in A050932. The e.g.f. for (n+1)*Bernoulli(n), n >= 0, is (d/dx)(x^2/(exp(x)-1)) = x*(2*(exp(x)-1)- x*exp(x))/(exp(x)-1)^2. - Wolfdieter Lang, Jul 15 2013
It can be observed that the rational sequence [0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, ...], derived from a(n)/A050932(n), is an autosequence of the first kind. - Jean-François Alcover, Jul 21 2017
Apparently a(n) = numerator(Sum_{k=0..n-1} (-1)^(n-k+1)*E1(n,k+1)/binomial(n,k+1)) for n >= 2, where E1(n, k) denotes the first-order Eulerian numbers A123125. - Peter Luschny, Feb 17 2021
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LINKS
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MATHEMATICA
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Numerator[Table[(n+1)BernoulliB[n], {n, 0, 40}]] (* Harvey P. Dale, May 13 2012 *)
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PROG
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(Haskell)
a050925 n = a050925_list !! n
a050925_list = 1 : -1 : (tail $ map (numerator . sum) $
zipWith (zipWith (%))
(zipWith (map . (*)) (drop 2 a000142_list) a242179_tabf) a106831_tabf)
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CROSSREFS
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KEYWORD
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sign,easy,frac,nice
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AUTHOR
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STATUS
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approved
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A106831
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Define a triangle in which the entries are of the form +-1/(b!c!d!e!...), where the order of the factorials is important; read the triangle by rows and record and expand the denominators.
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+10
9
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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, 288, 288, 192, 720, 288, 216, 144, 288, 144, 144, 96, 1440, 480, 288, 192, 288, 144, 144, 96, 480, 192, 144, 96, 192
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OFFSET
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0,1
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COMMENTS
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Row n has 2^n terms. Row 0 is +1/2!. An entry +-1/b!c!d!... has two children, a left child -+1/(a+1)!b!c!... and a right child +-1/2!b!c!d!...
Let S_n = sum of entries in row n of the triangle. Then for n > 0, n!S_{n-1} is the Bernoulli number B_n.
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LINKS
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FORMULA
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If sequence is shifted one term to the right, then the following recurrence works:
a(0) = 1; and for n > 0, a(2n) = (1+A001511(2n))*a(n), a(2n+1) = 2*a(n).
(End)
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EXAMPLE
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Woon's "Bernoulli Tree" begins like this (see also the given Wikipedia-link). This sequence gives the values of the denominators:
+1
────
2!
-1 / \ +1
──── ............../ \.............. ─────
3! 2!2!
+1 . -1 -1 . +1
──── / \ ──── ──── / \ ──────
4! ...../ \..... 2!3! 3!2! ...../ \.... 2!2!2!
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
-1 +1 +1 -1 +1 -1 -1 +1
──── ──── ──── ────── ──── ────── ────── ────────
5! 2!4! 3!3! 2!2!3! 4!2! 2!3!2! 3!2!2! 2!2!2!2!
etc.
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MAPLE
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The routine computes the triangle row by row and gives the numbers with their sign.
Thus A(1)=[2]; A(2)=[ -6, 4]; A(3)=[24, -12, -12, 8]; etc.
A := proc(n) local k, i, j, m, W, T; k := 2;
W := array(0..2^n); W[1] := [1, `if`(n=0, 1, 2)];
for i from 1 to n-1 do for m from k by 2 to 2*k-1 do
T := W[iquo(m, 2)]; W[m] := [ -T[1], T[2]+1, seq(T[j], j=3..nops(T))];
W[m+1] := [T[1], 2, seq(T[j], j=2..nops(T))]; od; k := 2*k; od;
seq(W[i][1]*mul(W[i][j]!, j=2..nops(W[i])), i=iquo(k, 2)..k-1) end:
seq(print(A(i)), i=1..5); (End)
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MATHEMATICA
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a [n_] := Module[{k, i, j, m, w, t}, k = 2; w = Array[0&, 2^n]; w[[1]] := {1, If[n == 0, 1, 2]}; For[i = 1, i <= n-1, i++, For[m = k, m <= 2*k-1 , m = m+2, t = w[[Quotient[m, 2]]]; w[[m]] = {-t[[1]], t[[2]]+1, Sequence @@ Table[t[[j]], {j, 3, Length[t]}]}; w[[m+1]] = {t[[1]], 2, Sequence @@ Table[t[[j]], {j, 2, Length[t]}]}]; k = 2*k]; Table[w[[i, 1]]*Product[w[[i, j]]!, {j, 2, Length[w[[i]]]}], {i, Quotient[k, 2], k-1}]]; Table[a[i] , {i, 1, 6}] // Flatten // Abs (* Jean-François Alcover, Dec 20 2013, translated from Maple *)
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PROG
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(Haskell)
a106831 n k = a106831_tabf !! n !! n
a106831_row n = a106831_tabf !! n
a106831_tabf = map (map (\(_, _, left, right) -> left * right)) $
iterate (concatMap (\(x, f, left, right) -> let f' = f * x in
[(x + 1, f', f', right), (3, 2, 2, left * right)])) [(3, 2, 2, 1)]
(PARI)
A106831off1(n) = if(!n, 1, my(rl=1, m=1); while(n, if(!(n%2), rl++, m *= ((1+rl)!); rl=1); n >>= 1); (m));
(PARI)
A106831r1(n) = if(!n, 1, if(n%2, 2*A106831r1((n-1)/2), (1+A001511(n))*A106831r1(n/2))); \\ Implements the given recurrence.
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CROSSREFS
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KEYWORD
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nonn,tabf,frac,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A233316
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a(n) = Numerator(binomial(n+2, 2)*Bernoulli(n, 1)) for n >= 0 and 0 for n < 0.
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+10
2
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0, 0, 1, 3, 1, 0, -1, 0, 2, 0, -3, 0, 5, 0, -691, 0, 140, 0, -10851, 0, 219335, 0, -1222277, 0, 1709026, 0, -1181820455, 0, 538845489, 0, -23749461029, 0, 68926730208040, 0, -84802531453387, 0, 270657225128535, 0, -26315271553053477373, 0, 380899208799402670, 0, -1827579029475143854357
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OFFSET
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-2,4
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COMMENTS
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Br2(n) = 0, 0, 1, 3/2, 1, 0, -1/2, 0, 2/3, 0, -3/2, 0, 5, 0, -691/30, ..., second complementary Bernoulli numbers.
Br2(n) differences table:
0, 0, 1, 3/2, 1, 0, -1/2, ...
0, 1, 1/2, -1/2, -1, -1/2, 1/2, ...
1, -1/2, -1, -1/2, 1/2, 1, 1/6, ...
-3/2, -1/2, 1/2, 1, 1/2, -5/6, -3/2, ...
1, 1, 1/2, -1/2, -4/3, -2/3, 2, ...
0, -1/2, -1, -5/6, 2/3, 8/3, 4/3, ...
-1/2, -1/2, 1/6, 3/2, 2, -4/3, -8, ... .
The main diagonal is the double of the first upper diagonal. Then, the autosequence (its inverse binomial transform is the signed sequence) is of second kind. Note that Br0(n) is an autosequence of second kind and Br(n) an autosequence of first kind.
First Bernoulli polynomials, i.e., for B(1) = -1/2, A196838/A196839, with 0's instead of the spaces:
1, 0, 0, 0, 0, 0, 0, 0, 0, ...
-1/2, 1, 0, 0, 0, 0, 0, 0, 0, ...
1/6, -1, 1, 0, 0, 0, 0, 0, 0, ...
0, 1/2, -3/2, 1, 0, 0, 0, 0, 0, ...
-1/30, 0, 1, -2, 1, 0, 0, 0, 0, ...
0, -1/6, 0, 5/3, -5/2, 1, 0, 0, 0, ...
1/42, 0, -1/2, 0, 5/2, -3, 1, 0, 0, ...
0, 1/6, 0, -7/6, 0, 0, -7/2, 1, 0, ...
-1/30, 0, 2/3, 0, -7/3, 0, 14/3, -4, 10, ... .
Second column: A229979/c(n) with -1 instead of 1, first column in A229979.
Third column: Br2(n) with -3/2 instead of 3/2, first column of the first array.
Etc.
Sequences used for Brp(n). For p=1, Br(n) is used.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ... = A001477,
0, 0, 1, 3, 6, 10, 15, 21, 28, 36, ...
0, 0, 0, 1, 4, 10, 20, 35, 56, 84, ... . See A052553.
For every sequence, the multiplication by A164555/A027642 begins at 1.
(Br0(n), Br(n), Br2(n), Br3(n), ... lead to A193815.)
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LINKS
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EXAMPLE
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a(2) = 1*1 = 1,
a(3) = 3*1/2 = 3/2,
a(4) = 6/6 = 1,
a(5) = 10*0 = 0,
a(6) = -15/30 = -1/2.
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MATHEMATICA
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b[-2] = b[-1] = 0; b[1] = 1/2; b[n_] := BernoulliB[n]; a[n_] := (n+1)*(n+2)/2*b[n] // Numerator; Table[a[n], {n, -2, 40}] (* Jean-François Alcover, Dec 09 2013 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A166123
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If n is prime, a(n) = 1; otherwise, a(n) is gcd(n, d) where d is the denominator of the (n-1)-th Bernoulli number.
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+10
1
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1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 1, 3, 1
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OFFSET
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1,9
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LINKS
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FORMULA
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MATHEMATICA
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Table[If[PrimeQ[n], 1, GCD[n, Denominator[BernoulliB[n-1]]]], {n, 100}] (* Harvey P. Dale, Sep 07 2017 *)
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PROG
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(PARI) a(n)=my(b=bernfrac(n-1)); denominator(b)/denominator(b*n)/if(isprime(n), n, 1) \\ Charles R Greathouse IV, Jun 20 2011
(PARI) a(n)=if(isprime(n), 1, my(b=bernfrac(n-1)); denominator(b)/denominator(b*n)) \\ Charles R Greathouse IV, Jun 20 2011
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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