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Search: id:A102756
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| A102756 |
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Triangle T(n,k), 0<=k<=n, read by rows defined by : T(n,k)=T(n-1,k-1)+2*T(n-1,k)+T(n-2,k-2)-T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or if n<k. |
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1
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| 1, 2, 1, 3, 4, 2, 4, 10, 10, 3, 5, 20, 31, 20, 5, 6, 35, 76, 78, 40, 8, 7, 56, 161, 232, 184, 76, 13, 8, 84, 308, 582, 636, 406, 142, 21, 9, 120, 546, 1296, 1831, 1604, 861, 260, 34, 10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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Rising and falling diagonals are A008999, A124400.
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FORMULA
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Sum_{k, 0<=k<=n}x^k*T(n,k)=A000012(n), A000027(n+1), A000244(n), A015530(n+1), A015544(n+1) for n=-1, 0, 1, 2, 3 . Sum_{k, 0<=k<=n}(-2)^k*T(n,k)= [1,0,3,0,9,0,27,0,81,0,...]for n=0,1,2,3,4,5,..., powers of 3 alternating with zeros . T(n,n-1)=2*A001629(n+1) for n>=1 . T(n,n)=Fibonacci(n+1)=A000045(n+1) . T(n,0)=n+1 . T(n,1)=A000292(n) for n>=1 . T(n+1,2)= binomial(n+4,n-1)+binomial(n+2,n-1)=A051747(n) for n>=1.
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EXAMPLE
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Triangle begins:
1;
2, 1;
3, 4, 2;
4, 10, 10, 3;
5, 20, 31, 20, 3;
6, 35, 76, 78, 40, 8;
7, 56, 161, 232, 184, 76, 13;
8, 84, 308, 582, 636, 406, 142, 21;
9, 120, 546, 1296, 1831, 1604, 861, 260, 34;
10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55;
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CROSSREFS
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Adjacent sequences: A102753 A102754 A102755 this_sequence A102757 A102758 A102759
Sequence in context: A125100 A128544 A120058 this_sequence A086614 A108959 A107893
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KEYWORD
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nonn,tabl
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AUTHOR
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Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 18 2006
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