1,2

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]

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

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]

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

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 ...

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

(PARI) T(n, k)=polcoeff((3+x)^n, k)

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: A132819 A105545 A164942 this_sequence A157393 A127552 A052931

Adjacent sequences: A027462 A027463 A027464 this_sequence A027466 A027467 A027468

nonn,tabl,easy,nice

Olivier Gerard (olivier.gerard(AT)gmail.com), N. J. A. Sloane (njas(AT)research.att.com).