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A116412
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Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
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2
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1, 3, 1, 6, 6, 1, 12, 21, 9, 1, 24, 60, 45, 12, 1, 48, 156, 171, 78, 15, 1, 96, 384, 558, 372, 120, 18, 1, 192, 912, 1656, 1473, 690, 171, 21, 1, 384, 2112, 4608, 5160, 3225, 1152, 231, 24, 1, 768, 4800, 12240, 16584, 13083, 6219, 1785, 300, 27, 1, 1536, 10752
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Row sums are A003688. Diagonal sums are A116413. Product of A007318 and A116413 is A116414. Product of A007318 and A105475.
Subtriangle of triangle given by (0, 3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - DELEHAM Philippe, Jan 18 2012
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LINKS
| Vladimir Kruchinin, Compositae and their properties, arXiv:1103.2582
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FORMULA
| Number triangle T(n,k)=sum{j=0..n, C(k+1,j)*C(n-j,k)2^(n-k-j)}
Contribution from Vladimir Kruchinin, Mar 17 2011: (Start)
T((m+1)*n+r-1, m*n+r-1) * r/(m*n+r) = sum(k=1..n, k/n * T((m+1)*n-k-1, m*n-1) * T(r+k-1,r-1)), n>=m>1.
T(n-1,m-1) = m/n * sum(k=1..n-m+1, k*A003945(k-1)*T(n-k-1,m-2)), n>=m>1. (End)
G.f.: (1+x)/(1-(y+2)*x -y*x^2). - DELEHAM Philippe, Jan 18 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A104537(n), A110523(n), (-2)^floor(n/2), A057079(n), A003945(n), A003688(n+1), A123347(n), A180035(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 repectively. - DELEHAM Philippe, Jan 18 2012
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EXAMPLE
| Triangle begins
1,
3, 1,
6, 6, 1,
12, 21, 9, 1,
24, 60, 45, 12, 1,
48, 156, 171, 78, 15, 1
Triangle T(n,k), 0<=k<=n, given by (0, 3, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins :
1
0, 1
0, 3, 1
0, 6, 6, 1
0, 12, 21, 9, 1
0, 24, 60, 45, 12, 1
0, 48, 156, 171, 78, 15, 1
... - DELEHAM Philippe, Jan 18 2012
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CROSSREFS
| Cf. A003688, A003945
Sequence in context: A127893 A127895 A152685 * A089511 A112692 A198614
Adjacent sequences: A116409 A116410 A116411 * A116413 A116414 A116415
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Feb 13 2006
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