This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A027465 #71 Oct 09 2023 01:44:34
%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 Rows of A013610 reversed. - _Michael Somos_, Feb 14 2002
%C A027465 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
%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_, Mar 17 2006
%C A027465 Mirror image of A013610. - _Zerinvary Lajos_, 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_, Dec 21 2007
%C A027465 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
%C A027465 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
%C A027465 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
%C A027465 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
%C A027465 Also the convolution triangle of A000244. - _Peter Luschny_, Oct 09 2022
%H A027465 Reinhard Zumkeller, <a href="/A027465/b027465.txt">Rows n = 0..125 of triangle, flattened</a>
%H A027465 B. N. Cyvin et al., <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match34/match34_109-121.pdf">Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons</a>, Match, No. 34 (Oct 1996), pp. 109-121.
%H A027465 Erich Neuwirth, <a href="https://doi.org/10.1016/S0012-365X(00)00373-3">Recursively defined combinatorial functions: Extending Galton's board</a>, Discrete Math. 239 No. 1-3, 33-51 (2001).
%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_{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
%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_, 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. - _Tom Copeland_, Aug 18 2008
%F A027465 T(n,k) = A164942(n,k)*(-1)^k. - _Philippe Deléham_, Oct 09 2011
%F A027465 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,
%F A027465 exp[x*z/(1-3z)]/(1-3z) = exp(3z D_z z) e^(x*z)= exp(3D_x x D_x) e^(z*x)
%F A027465 = (1 z z^2 z^3 ...) H (1 x x^2/2! x^3/3! ...)^T
%F A027465 = (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
%F A027465 E.g.f. for column k: x^k/k! * exp(3x). - _Geoffrey Critzer_, Sep 04 2013
%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 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}.
%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_, Nov 25 2007
%p A027465 # Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
%p A027465 PMatrix(10, n -> 3^(n-1)); # _Peter Luschny_, Oct 09 2022
%t A027465 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 *)
%o A027465 (PARI) {T(n, k) = polcoeff( (3 + x)^n, k)}; /* _Michael Somos_, Feb 14 2002 */
%o A027465 (Haskell)
%o A027465 a027465 n k = a027465_tabl !! n !! k
%o A027465 a027465_row n = a027465_tabl !! n
%o A027465 a027465_tabl = iterate (\row ->
%o A027465 zipWith (+) (map (* 3) (row ++ [0])) (map (* 1) ([0] ++ row))) [1]
%o A027465 -- _Reinhard Zumkeller_, May 26 2013
%Y A027465 Cf. A000244, A007318, A013610, A013610, A099097, A027471, A027472, A036216, A036217, A036219, A036220, A036221, A036222, A036223.
%K A027465 nonn,tabl,easy,nice
%O A027465 0,2
%A A027465 _Olivier Gérard_, _N. J. A. Sloane_