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 A027465 Cube of lower triangular normalized binomial matrix. 36
 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 Also the convolution triangle of A000244. - Peter Luschny, Oct 09 2022 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 < m. G.f. for m-th column: (x/(1-3*x))^m (m-fold convolution of A000244, powers of 3). - Wolfdieter Lang, Feb 2006 G.f.: 1 / (1 - x(3+y)). a(n,k) = 3*a(n-1,k) + a(n-1,k-1) - R. J. Mathar, 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. - Tom Copeland, Aug 18 2008 T(n,k) = A164942(n,k)*(-1)^k. - Philippe Deléham, Oct 09 2011 Let P and P^T be the Pascal matrix and its transpose and H = P^3 = A027465. Then from the formalism of A132440 and A218272, exp[x*z/(1-3z)]/(1-3z) = exp(3z D_z z) e^(x*z)= exp(3D_x x D_x) e^(z*x) = (1 z z^2 z^3 ...) H (1 x x^2/2! x^3/3! ...)^T = (1 x x^2/2! x^3/3! ...) H^T (1 z z^2 z^3 ...)^T = Sum_{n>=0} (3z)^n L_n(-x/3), where D is the derivative operator and L_n(x) are the regular (not normalized) Laguerre polynomials. - Tom Copeland, Oct 26 2012 E.g.f. for column k: x^k/k! * exp(3x). - Geoffrey Critzer, Sep 04 2013 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 ... MAPLE for i from 0 to 12 do seq(binomial(i, j)*3^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Nov 25 2007 # Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left. PMatrix(10, n -> 3^(n-1)); # Peter Luschny, Oct 09 2022 MATHEMATICA t[n_, k_] := Binomial[n, k]*3^(n-k); Table[t[n, n-k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 19 2012 *) PROG (PARI) {T(n, k) = polcoeff( (3 + x)^n, k)}; /* Michael Somos, Feb 14 2002 */ (Haskell) a027465 n k = a027465_tabl !! n !! k a027465_row n = a027465_tabl !! n a027465_tabl = iterate (\row -> zipWith (+) (map (* 3) (row ++ [0])) (map (* 1) ([0] ++ row))) [1] -- Reinhard Zumkeller, May 26 2013 CROSSREFS Cf. A000244, A007318, A013610, A013610, A099097, A027471, A027472, A036216, A036217, A036219, A036220, A036221, A036222, A036223. Sequence in context: A330509 A105545 A178831 * A164942 A236420 A187537 Adjacent sequences: A027462 A027463 A027464 * A027466 A027467 A027468 KEYWORD nonn,tabl,easy,nice AUTHOR Olivier Gérard, N. J. A. Sloane STATUS approved

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Last modified December 9 09:12 EST 2023. Contains 367690 sequences. (Running on oeis4.)