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A121408
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Triangle T(n,k) defined by the generating function (in Maple notation): exp(y*arcsin(x))-1 = sum( sum(T(n,k)*y^k, k=1..n)*x^n/n!, n=1..infinity).
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2
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1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 9, 0, 10, 0, 1, 0, 64, 0, 20, 0, 1, 225, 0, 259, 0, 35, 0, 1, 0, 2304, 0, 784, 0, 56, 0, 1, 11025, 0, 12916, 0, 1974, 0, 84, 0, 1, 0, 147456, 0, 52480, 0, 4368, 0, 120, 0, 1, 893025, 0, 1057221, 0, 172810, 0, 8778, 0, 165, 0, 1, 0, 14745600, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| Row sums are equal to A006228(n). This is sequence A091885 with additional intertwining zeros.
F(n,m) = n!*T(n,m)/m! is a composite (akin to Riordan arrays) of F(x)=arcsin(x) and (F(x))^m = sum{n=m..infinity} F(n,m)*x^n, and for o.g.f. G(x), G(arcsin(x)) = g(0) +sum_{n=1..infinity} sum_{m=1..n} F(n,m)*g(m)*x^n, see the preprint. - Vladimir Kruchinin, Feb 10 2011
The unsigned matrix inverse is A136630 (with a different offset) - Peter
Bala, Feb 23 2011.
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LINKS
| Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565
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FORMULA
| T(n,m)= ((n-1)!/(m-1)!) *sum_{k=1..n-m} sum_{j=1..k} binomial(k,j) *(2^(1-j) /(n-m+j)!) *sum{i=0..floor(j/2)} (-1)^((n-m)/2-i-j) *binomial(j,i) *(j-2*i)^(n-m+j) *binomial(k+n-1,n-1), n>m and even(n-m). [From Vladimir Kruchinin, Feb 10 2011]
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EXAMPLE
| Triangle starts:
1;
0,1;
1,0,1;
0,4,0,1;
9,0,10,0,1;
0,64,0,20,0,1;
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MAPLE
| g:=exp(y*arcsin(x))-1: gser:=simplify(series(g, x=0, 15)): for n from 1 to 12 do P[n]:=sort(n!*coeff(gser, x, n)) od: for n from 1 to 12 do seq(coeff(P[n], y, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A006228, A091885, A136630.
Sequence in context: A201560 A059678 A079642 * A186761 A199786 A189245
Adjacent sequences: A121405 A121406 A121407 * A121409 A121410 A121411
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 28 2006
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