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A000737
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Boustrophedon transform of natural numbers, cf. A000027.
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5
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1, 3, 8, 21, 60, 197, 756, 3367, 17136, 98153, 624804, 4375283, 33424512, 276622829, 2465449252, 23543304919, 239810132288, 2595353815825, 29740563986500, 359735190398875, 4580290700420064, 61233976084442741
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
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FORMULA
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E.g.f.: (1 + x)*(tan x + sec x)*exp(x).
a(n) ~ n! * (Pi + 2)*exp(Pi/2)*2^(n+1)/Pi^(n+1). - Vaclav Kotesovec, Oct 02 2013
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MATHEMATICA
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CoefficientList[Series[(1+x)*(Tan[x]+1/Cos[x])* E^x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)
t[n_, 0] := n + 1; t[n_, k_] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)
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PROG
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(Sage) # Algorithm of L. Seidel (1877)
R = []; A = {-1:0, 0:0}
k = 0; e = 1
for i in range(n) :
Am = i+1
A[k + e] = 0
e = -e
for j in (0..i) :
Am += A[k]
A[k] = Am
k += e
# To trace the algorithm remove the comment sign.
# print([A[z] for z in (-i//2..i//2)])
R.append(A[e*i//2])
return R
(Haskell)
a000737 n = sum $ zipWith (*) (a109449_row n) [1..]
(Python)
from itertools import count, accumulate, islice
def A000737_gen(): # generator of terms
blist = tuple()
for i in count(1):
yield (blist := tuple(accumulate(reversed(blist), initial=i)))[-1]
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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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