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A105477
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Triangle read by rows: T(n,k) is the number of compositions of n into k parts when there are two kinds of part 2.
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1
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1, 2, 1, 1, 4, 1, 1, 6, 6, 1, 1, 6, 15, 8, 1, 1, 7, 23, 28, 10, 1, 1, 8, 30, 60, 45, 12, 1, 1, 9, 39, 98, 125, 66, 14, 1, 1, 10, 49, 144, 255, 226, 91, 16, 1, 1, 11, 60, 202, 437, 561, 371, 120, 18, 1, 1, 12, 72, 272, 685, 1128, 1092, 568, 153, 20, 1, 1, 13, 85, 355, 1015, 1995, 2555
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Triangle T(n,k), 1<=k<=n, given by (0, 2, -3/2, -1/6, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. Triangle T(n,k), 0<=k<=n, is the Riordan array (1, x*(1+x-x^2)/(1-x)) . - DELEHAM Philippe, Jan 25 2012
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FORMULA
| G.f.=tz(1+z-z^2)/(1-z-tz-tz^2+tz^3).
T(n,k)=Sum(binomial(k,j)*binomial(n-2j-1, k-j-1), j=0..n-k). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 06 2006
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) - T(n-3,k-1), n>1. - DELEHAM Philippe, Jan 25 2012
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EXAMPLE
| T(4,2)=6 because we have (1,3),(3,1),(2,2),(2,2'),(2',2) and (2',2').
Triangle begins:
1;
2,1;
1,4,1;
1,6,6,1;
1,6,15,8,1;
Triangle T(n,k) given by (0,2,-3/2,-1/6,2/3,0,0,0,...) DELTA (1,0,0,0,0,...) begins :
1
0, 1
0, 2, 1
0, 1, 4, 1
0, 1, 6, 6, 1
0, 1, 6, 15, 8, 1...
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MAPLE
| G:=t*z*(1+z-z^2)/(1-z-t*z-t*z^2+t*z^3): Gser:=simplify(series(G, z=0, 15)): for n from 1 to 14 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 13 do seq(coeff(P[n], t^k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
| Row sums yield A077998.
Diagonals : A000012, A005843, A000384
Sequence in context: A061462 A122578 A005131 * A127709 A199856 A193554
Adjacent sequences: A105474 A105475 A105476 * A105478 A105479 A105480
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005
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