A309931 - OEIS

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A309931

Triangle read by rows: T(n,k) is the number of compositions of n with k parts and circular differences all equal to 1, 0, or -1.

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 3, 4, 1, 1, 1, 1, 6, 5, 1, 1, 2, 3, 4, 10, 6, 1, 1, 1, 3, 5, 10, 15, 7, 1, 1, 2, 1, 4, 10, 20, 21, 8, 1, 1, 1, 3, 6, 11, 21, 35, 28, 9, 1, 1, 2, 3, 4, 10, 24, 42, 56, 36, 10, 1, 1, 1, 1, 5, 10, 25, 49, 78, 84, 45, 11, 1

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OFFSET

1,5

LINKS

Table of n, a(n) for n=1..78.

FORMULA

T(n, 1) = T(n, n) = 1.

T(n, 2) = (3 - (-1)^n)/2.

T(n, n - 1) = binomial(n-1, 1) = n - 1.

T(n, n - 2) = binomial(n-2, 2).

EXAMPLE

Triangle begins:

1, 1;

1, 2, 1;

1, 1, 3, 1;

1, 2, 3, 4, 1;

1, 1, 1, 6, 5, 1;

1, 2, 3, 4, 10, 6, 1;

1, 1, 3, 5, 10, 15, 7, 1;

1, 2, 1, 4, 10, 20, 21, 8, 1;

1, 1, 3, 6, 11, 21, 35, 28, 9, 1;

1, 2, 3, 4, 10, 24, 42, 56, 36, 10, 1;

1, 1, 1, 5, 10, 25, 49, 78, 84, 45, 11, 1;

1, 2, 3, 4, 10, 24, 56, 96, 135, 120, 55, 12, 1;

1, 1, 3, 6, 10, 21, 57, 116, 180, 220, 165, 66, 13, 1;

...

For n = 6 there are a total of 15 compositions:

k = 1: (6)

k = 2: (33)

k = 3: (222)

k = 4: (1122), (1212), (1221), (2112), (2121), (2211)

k = 5: (11112), (11121), (11211), (12111), (21111)

k = 6: (111111)

PROG

(PARI)

step(R, n)={matrix(n, n, i, j, if(i>j, if(j>1, R[i-j, j-1]) + R[i-j, j] + if(j+1<=n, R[i-j, j+1])) )}

T(n)={my(v=vector(n)); for(k=1, n, my(R=matrix(n, n, i, j, i==j&&abs(i-k)<=1), m=0); while(R, m++; v[m]+=R[n, k]; R=step(R, n))); v}

for(n=1, 12, print(T(n)));

CROSSREFS

Row sums are A325591.

Cf. A309937, A309938, A309939.

Sequence in context: A101950 A104562 A164306 * A309939 A111603 A180178

Adjacent sequences: A309928 A309929 A309930 * A309932 A309933 A309934

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Aug 23 2019

STATUS

approved