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Search: id:A092689
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| A092689 |
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Triangle, read by rows, such that the convolution of each row with {1,2} produces a triangle which, after the main diagonal is divided by 2, and the triangle is flattened, equals this flattened form of the original triangle. |
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8
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| 1, 1, 1, 3, 1, 3, 7, 5, 3, 7, 19, 13, 13, 7, 19, 51, 39, 33, 33, 19, 51, 141, 111, 99, 85, 89, 51, 141, 393, 321, 283, 259, 229, 243, 141, 393, 1107, 925, 825, 747, 701, 627, 675, 393, 1107, 3139, 2675, 2397, 2195, 2029, 1929, 1743, 1893, 1107, 3139, 8953, 7747
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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First column and main diagonal forms the central trinomial coefficients (A002426). Row sums form A092690.
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FORMULA
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T(n, k) = 2*T(n-1, k) + T(n-1, k+1) for 0<k<n, with T(n, n)=T(n, 0)=T(n+1, n)=A002426(n), T(0, 0)=1, T(0, 1)=T(1, 0)=1.
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EXAMPLE
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Rows begin:
{1},
{1,1},
{3,1,3},
{7,5,3,7},
{19,13,13,7,19},
{51,39,33,33,19,51},
{141,111,99,85,89,51,141},
{393,321,283,259,229,243,141,393},
{1107,925,825,747,701,627,675,393,1107},
{3139,2675,2397,2195,2029,1929,1743,1893,1107,3139},
{8953,7747,6989,6419,5987,5601,5379,4893,5353,3139,8953},...
Convolution of each row with {1,2} forms the triangle:
{1,2},
{1,3,2},
{3,7,5,6},
{7,19,13,13,14},
{19,51,39,33,33,38},
{51,141,111,99,85,89,102},
{141,393,321,283,259,229,243,282},...
which, after the main diagonal is divided by 2 and the triangle is flattened, equals the original triangle in flattened form: {1,1,1,3,1,3,7,5,3,7,19,...}.
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PROGRAM
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(PARI) {T(n, k)=if(n<0|k>n, 0, if(n==0&k==0, 1, if(n==1&k<=1, 1, if(k==n-1, T(n-1, 0), if(k==n, T(n, 0), 2*T(n-1, k)+T(n-1, k+1))))))}
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CROSSREFS
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Cf. A002426, A092683, A092686, A092690.
Sequence in context: A114972 A107461 A035619 this_sequence A064434 A086401 A095732
Adjacent sequences: A092686 A092687 A092688 this_sequence A092690 A092691 A092692
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Mar 04 2004
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