A092689 - OEIS

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A092689

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.

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; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

First column and main diagonal forms the central trinomial coefficients (A002426). Row sums form A092690.

LINKS

Table of n, a(n) for n=0..56.

FORMULA

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.

EXAMPLE

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,...}.

PROG

(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))))))

CROSSREFS

Cf. A002426, A092683, A092686, A092690.

Sequence in context: A107461 A035619 A280995 * A281553 A064434 A328988

Adjacent sequences: A092686 A092687 A092688 * A092690 A092691 A092692

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Mar 04 2004

STATUS

approved