%I A027465
%S A027465 1,3,1,9,6,1,27,27,9,1,81,108,54,12,1,243,405,270,90,15,1,729,1458,
%T A027465 1215,540,135,18,1,2187,5103,5103,2835,945,189,21,1,6561,17496,20412,
%U A027465 13608,5670,1512,252,24,1,19683,59049,78732,61236,30618,10206,2268
%N A027465 Cube of lower triangular normalized binomial matrix.
%C A027465 Row sums are powers of 4 (A000302), antidiagonal sums are A006190 (a(n) 
               = 3*a(n-1) + a(n-2)). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), 
               May 17 2005
%C A027465 Triangle of coefficients in expansion of (3+x)^n.
%C A027465 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 
               (mathar(AT)strw.leidenuniv.nl), Mar 17 2006
%C A027465 Mirror image of A013610. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 25 2007
%C A027465 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 (zerinvarylajos(AT)yahoo.com), 
               Dec 21 2007
%C A027465 Anti-diagonals square array give bisection Fibonacci sequence: A001906. 
               Example: 81-(27-1)=55. Differential rule applied to rows give A000079. 
               [From M. Dols (markdols99(AT)yahoo.com), Sep 01 2009]
%D A027465 B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal 
               systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 
               109-121.
%D A027465 E. Neuwirth, Recursively defined combinatorial functions: extending Galton's 
               board, Disc. Math 239 (2001) 33-51
%F A027465 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.
%F A027465 Triangle whose (i, j)-th entry is binomial(i, j)*3^(i-j).
%F A027465 a(n, m)= 4^(n-1)*sum(b(n, j)*b(j, m), j=m..n)= 3^(n-m)*binomial(n-1, 
               m-1), n >= m >= 1; a(n, m) := 0, n<m. G.f. for m-th column: (x/(1-3*x))^m 
               (m-fold convolution of A000244, powers of 3) - from Wolfdieter Lang 
               (wolfdieter.lang(AT)physik.uni-karlsruhe.de).
%F A027465 G.f.: 1 / [1 - x(3+y)]
%F A027465 a(n,k)=3*a(n-1,k)+a(n-1,k-1) - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 17 2006
%F A027465 From the formalism of A133314, the e.g.f. for the row polynomials of 
               A027465 is exp(x*t)*exp(3x). The e.g.f. for the row polynomials of 
               the inverse matrix is exp(x*t)*exp(-3x). p iterates of the matrix 
               give the matrix with e.g.f. exp(x*t)*exp(p*3x). The results generalize 
               for 3 replaced by any number. [From Tom Copeland (tcjpn(AT)msn.com), 
               Aug 18 2008]
%e A027465 Example: n = 3 offers 2^3 = 8 different binary vectors (0,0,0),(0,0,1),
               ...(1,1,0), (1,1,1). a(3,2) = 9 of the 2^4 = 64 pairs have overlap 
               k = 2: (0,1,1)*(0,1,1) = (1,0,1)*(1,0,1) = (1,1,0)*(1,1,0) = (1,1,
               1)*(1,1,0) = (1,1,1)*(1,0,1) = (1,1,1)*(0,1,1) = (0,1,1)*(1,1,1) 
               = (1,0,1)*(1,1,1) = (1,1,0)*(1,1,1) = 2
%e A027465 The present sequence formatted as a triangular array:
%e A027465 1
%e A027465 3 1
%e A027465 9 6 1
%e A027465 27 27 9 1
%e A027465 81 108 54 12 1
%e A027465 243 405 270 90 15 1
%e A027465 729 1458 1215 540 135 18 1
%e A027465 2187 5103 5103 2835 945 189 21 1
%e A027465 6561 17496 20412 13608 5670 1512 252 24 1
%e A027465 ...
%e A027465 A013610 formatted as a triangular array:
%e A027465 1
%e A027465 1 3
%e A027465 1 6 9
%e A027465 1 9 27 27
%e A027465 1 12 54 108 81
%e A027465 1 15 90 270 405 243
%e A027465 1 18 135 540 1215 1458 729
%e A027465 1 21 189 945 2835 5103 5103 2187
%e A027465 1 24 252 1512 5670 13608 20412 17496 6561
%e A027465 ...
%e A027465 A099097 formatted as a square array:
%e A027465 1 0 0 0 0 0 0 0 0 0 0 ...
%e A027465 3 1 0 0 0 0 0 0 0 0 ...
%e A027465 9 6 1 0 0 0 0 0 0 ...
%e A027465 27 27 9 1 0 0 0 0 ...
%e A027465 81 108 54 12 1 0 0 ...
%e A027465 243 405 270 90 15 1 ...
%e A027465 729 1458 1215 540 135 ...
%e A027465 2187 5103 5103 2835 ...
%e A027465 6561 17496 20412 ...
%e A027465 19683 59049 ...
%e A027465 59049 ...
%p A027465 for i from 0 to 12 do seq(binomial(i, j)*3^(i-j), j = 0 .. i) od; - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Nov 25 2007
%o A027465 (PARI) T(n,k)=polcoeff((3+x)^n,k)
%Y A027465 The rows of A013610 are the rows of A027465 reversed.
%Y A027465 Cf. A007318, A013610.
%Y A027465 Cf. A013610 A099097 A000244, A027471, A027472, A036216, A036217, A036219, 
               A036220, A036221, A036222, A036223.
%Y A027465 Sequence in context: A132819 A105545 A164942 this_sequence A157393 A127552 
               A052931
%Y A027465 Adjacent sequences: A027462 A027463 A027464 this_sequence A027466 A027467 
               A027468
%K A027465 nonn,tabl,easy,nice
%O A027465 1,2
%A A027465 Olivier Gerard (olivier.gerard(AT)gmail.com), N. J. A. Sloane (njas(AT)research.att.com).