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Search: id:A051159
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| A051159 |
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Triangular array made of three copies of Pascal's triangle. |
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10
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| 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 3, 0, 3, 0, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1, 1, 1, 4, 4, 6, 6, 4, 4, 1, 1, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 1, 1, 5, 5, 10, 10, 10, 10, 5, 5, 1, 1, 1, 0, 6, 0, 15, 0, 20, 0, 15, 0, 6, 0, 1, 1, 1, 6, 6, 15, 15
(list; table; graph; listen)
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OFFSET
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0,13
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COMMENT
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Computing each term modulo 2 also gives A047999, i.e. A051159[ n ] mod 2 = A007318[ n ] mod 2 for all n. (The triangle is paritywise isomorphic to Pascal's Triangle) - Antti Karttunen
5th row/column gives entries of A000217 (triangular numbers C(n+1,2)) repeated twice and every other entry in 6th row/column form A000217. 7th row/column gives entries of A000292 (Tetrahedral (or pyramidal) nos: C(n+3,3)) repeated twice and every other entry in 8th row/column form A000292. 9th row/column gives entries of A000332 (binomial coefficients binomial(n,4)) repeated twice and every other entry in 10th row/column form A000332. 11th row/column gives entries of A000389 (binomial coefficients C(n,5)) repeated twice and every other entry in 12th row/column form A000389. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 21 2004
If Sum_{k=0..n}A(k)*T(n,k)=B(n), the sequence B is the S-D transform of the sequence A . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 02 2006
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REFERENCES
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S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
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FORMULA
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T(n, k)=T(n-1, k-1)+T(n-1, k) if n odd or k even, else 0. T(0, 0)=1.
T(n, k)=T(n-2, k-2)+T(n-2, k). T(0, 0)=T(1, 0)=T(1, 1)=1.
Square array made by setting first row/column to 1's (A(i, 0) = A(0, j) = 1); A(1, 1) = 0; A(1, j) = A(1, j-2); A(i, 1) = A(i-2, 1); other entries A(i, j) = A(i-2, j) + A(i, j-2). - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 21 2004
Sum_{k=0..n}k*T(n,k)=A093968(n); A093968 = S-D transform of A001477 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 02 2006
Equals 2*A007318 - A034851 (twice Pascal's triangle - the Losanitch triangle). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 31 2007
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EXAMPLE
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1; 1 1; 1 0 1; 1 1 1 1; 1 0 2 0 1; 1 1 2 2 1 1; ...
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PROGRAM
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(PARI) T(n, k)=C(n%2, k%2)*C(n\2, k\2); C(n, k)=if(k<0|k>n, 0, n!/k!/(n-k)!)
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CROSSREFS
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Cf. A007318. A051160(n, k)=(-1)^[ k/2 ]*A051159(n, k).
Cf. A034851.
Sequence in context: A029402 A035196 A051160 this_sequence A035697 A135549 A124737
Adjacent sequences: A051156 A051157 A051158 this_sequence A051160 A051161 A051162
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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Michael Somos
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