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A027465
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Cube of lower triangular normalized binomial matrix.
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25
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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
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 (mathar(AT)strw.leidenuniv.nl), Mar 17 2006
Mirror image of A013610. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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 (zerinvarylajos(AT)yahoo.com), Dec 21 2007
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]
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. - From DELEHAM Philippe, Oct 09 2011.
T(n,k)=C(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). [From Dennis P. Walsh, Feb 02 2012]
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REFERENCES
| B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
E. Neuwirth, Recursively defined combinatorial functions: extending Galton's board, Disc. Math 239 (2001) 33-51
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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(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).
G.f.: 1 / [1 - x(3+y)]
a(n,k)=3*a(n-1,k)+a(n-1,k-1) - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 17 2006
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]
T(n,k)=A164942(n,k)*(-1)^k. - From DELEHAM Philippe, Oct 09 2011.
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EXAMPLE
| 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
For example, T(2,1)=6 since there are 6 subsets of {1,2,3,4} that have exactly 1 symmetric pair, namely, {1,4}, {2,3}, {1,2,3}, {1,2,4}, {1,3,4}, and {2,3,4}.
The present sequence formatted as a triangular array:
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
...
A013610 formatted as a triangular array:
1
1 3
1 6 9
1 9 27 27
1 12 54 108 81
1 15 90 270 405 243
1 18 135 540 1215 1458 729
1 21 189 945 2835 5103 5103 2187
1 24 252 1512 5670 13608 20412 17496 6561
...
A099097 formatted as a square array:
1 0 0 0 0 0 0 0 0 0 0 ...
3 1 0 0 0 0 0 0 0 0 ...
9 6 1 0 0 0 0 0 0 ...
27 27 9 1 0 0 0 0 ...
81 108 54 12 1 0 0 ...
243 405 270 90 15 1 ...
729 1458 1215 540 135 ...
2187 5103 5103 2835 ...
6561 17496 20412 ...
19683 59049 ...
59049 ...
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MAPLE
| 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
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PROG
| (PARI) T(n, k)=polcoeff((3+x)^n, k)
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CROSSREFS
| The rows of A013610 are the rows of A027465 reversed.
Cf. A007318, A013610.
Cf. A013610 A099097 A000244, A027471, A027472, A036216, A036217, A036219, A036220, A036221, A036222, A036223.
Sequence in context: A105545 A178831 A164942 * A187537 A157393 A127552
Adjacent sequences: A027462 A027463 A027464 * A027466 A027467 A027468
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KEYWORD
| nonn,tabl,easy,nice,changed
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com), N. J. A. Sloane (njas(AT)research.att.com).
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