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A147308
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Riordan array [sech(x), asin(tanh(x))].
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3
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1, 0, 1, -1, 0, 1, 0, -4, 0, 1, 5, 0, -10, 0, 1, 0, 40, 0, -20, 0, 1, -61, 0, 175, 0, -35, 0, 1, 0, -768, 0, 560, 0, -56, 0, 1, 1385, 0, -4996, 0, 1470, 0, -84, 0, 1, 0, 24320, 0, -22720, 0, 3360, 0, -120, 0, 1, -50521, 0, 214445, 0
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OFFSET
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0,8
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COMMENTS
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Production array is [cos(x),x] beheaded. Inverse is A147309. Row sums are A012123(n+1).
If signs are ignored this is identical to A147309. - N. J. A. Sloane, Nov 07 2008
[gd(x)]^m=sum(n>=m T(n,m)*m!/n!*x^n), where gd(x) is Gudermannian function, T(n,m)=sum(j=0..(n-m)/2, (sum(i=0..2*j, (2^(i)*stirling1(i+m,m)*binomial(2*j+m-1,i+m-1))/(i+m)!))*sum(k=0..n-2*j-m, (-1)^(k+j)*binomial(k+2*j+m-1,2*j+m-1)*(k+2*j+m)!*2^(-k-2*j)*stirling2(n,k+2*j+m))), n>=m>=1. [From Vladimir Kruchinin, Dec 18 2011]
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LINKS
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Nathaniel Johnston, Table of n, a(n) for n = 0..229
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EXAMPLE
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Triangle begins
1,
0, 1,
-1, 0, 1,
0, -4, 0, 1,
5, 0, -10, 0, 1,
0, 40, 0, -20, 0, 1,
-61, 0, 175, 0, -35, 0, 1
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MAPLE
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Z := proc(n, x) option remember; if n = 0 then return 1: else return 1/2*x*(Z(n-1, x-1)+Z(n-1, x+1)): fi:end:
with(PolynomialTools): for n from 1 to 10 do for k from 1 to n do printf("%d, ", (-1)^floor((n-k)/2)*coeff(Z(n, x), x, k)):od: printf("\n"):od: # Nathaniel Johnston, Apr 21 2011
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CROSSREFS
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Sequence in context: A065623 A178103 A147309 * A110064 A021253 A136586
Adjacent sequences: A147305 A147306 A147307 * A147309 A147310 A147311
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KEYWORD
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easy,sign,tabl
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AUTHOR
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Paul Barry, Nov 05 2008
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STATUS
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approved
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