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 A027465 Cube of lower triangular normalized binomial matrix. 33
 1, 3, 1, 9, 6, 1, 27, 27, 9, 1, 81, 108, 54, 12, 1, 243, 405, 270, 90, 15, 1, 729, 1458, 1215, 540, 135, 18, 1, 2187, 5103, 5103, 2835, 945, 189, 21, 1, 6561, 17496, 20412, 13608, 5670, 1512, 252, 24, 1, 19683, 59049, 78732, 61236, 30618, 10206, 2268 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Rows of A013610 reversed. - Michael Somos, Feb 14 2002 Row sums are powers of 4 (A000302), antidiagonal sums are A006190 (a(n) = 3*a(n-1) + a(n-2)). - Gerald McGarvey, May 17 2005 Triangle of coefficients in expansion of (3+x)^n. Also: Pure Galton board of scheme (3,1). Also: Multiplicity (number) of pairs of n-dimensional binary vectors with dot product (overlap) k. There are 2^n = A000079(n) binary vectors of length n and 2^(2n) = 4^n = A000302(n) different pairs to form dot products k = Sum_{i=1..n}v[i]*u[i] between these, 0<=k<=n. (Since dot products are symmetric, there are only 2^n(2^n-1)/2 different non-ordered pairs, actually.) - R. J. Mathar, Mar 17 2006 Mirror image of A013610. - Zerinvary Lajos, Nov 25 2007 T(i,j) is the number of i-permutations of 4 objects a,b,c,d, with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007 The antidiagonals of the sequence formatted as a square array (see Examples section) and summed with alternating signs gives a bisection of Fibonacci sequence, A001906. Example: 81-(27-1)=55. Similar rule applied to rows gives A000079. - Mark Dols, Sep 01 2009 Triangle T(n,k), read by rows, given by (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. - Philippe Deléham, Oct 09 2011 T(n,k) = binomial(n,k)*3^(n-k), the number of subsets of [2n] with exactly k symmetric pairs, where elements i and j of [2n] form a symmetric pair if i+j=2n+1. Equivalently, if n couples attend a (ticketed) event that offers door prizes, then the number of possible prize distributions that have exactly k couples as dual winners is T(n,k). - Dennis P. Walsh, Feb 02 2012 T(n,k) is the number of ordered pairs (A,B) of subsets of {1,2,...,n} such that the intersection of A and B contains exactly k elements. For example, T(2,1) = 6 because we have ({1},{1}); ({1},{1,2}); ({2},{2}); ({2},{1,2}); ({1,2},{1}); ({1,2},{2}). Sum_{k=0..n} T(n,k)*k = A002697(n) (see comment there by Ross La Haye). - Geoffrey Critzer, Sep 04 2013 LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121. Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001) FORMULA Numerators of lower triangle of (b^2)[ i, j ] where b[ i, j ] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i. Triangle whose (i, j)-th entry is binomial(i, j)*3^(i-j). a(n, m) = 4^(n-1)*Sum_{j=m..n} b(n, j)*b(j, m) = 3^(n-m)*binomial(n-1, m-1), n >= m >= 1; a(n, m) := 0, n    zipWith (+) (map (* 3) (row ++ )) (map (* 1) ( ++ row)))  -- Reinhard Zumkeller, May 26 2013 CROSSREFS Cf. A007318, A013610. Cf. A013610 A099097 A000244, A027471, A027472, A036216, A036217, A036219, A036220, A036221, A036222, A036223. Sequence in context: A105545 A178831 A164942 * A236420 A187537 A246256 Adjacent sequences:  A027462 A027463 A027464 * A027466 A027467 A027468 KEYWORD nonn,tabl,easy,nice AUTHOR STATUS approved

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Last modified July 20 22:20 EDT 2019. Contains 325189 sequences. (Running on oeis4.)