A187055 Triangle T(n,r), read by rows, where the r-th column is expansion of A(x)^r, with A(x) = x * (x+1) * (2*x^4+4*x^3-2*x+1) * (x^4+2*x^3-x+1) / (x^2+x-1)^6.

1, 4, 1, 14, 8, 1, 46, 44, 12, 1, 141, 204, 90, 16, 1, 409, 846, 538, 152, 20, 1, 1132, 3234, 2787, 1112, 230, 24, 1, 3011, 11600, 13035, 6892, 1990, 324, 28, 1, 7736, 39502, 56372, 37956, 14345, 3236, 434, 32, 1, 19275, 128765, 228921, 191008, 90749, 26586, 4914, 560, 36, 1, 46724, 404228, 882291, 894364, 519580, 189798, 45311, 7088, 702, 40, 1

Offset

1,2

Comments

f(x) = (x+x^2)/(1-x-x^2), g(x)=x+x^2, h(x)=x+x^2+x^3, A(x)=g(h(f(x))).

A(x)^r = sum(n>=r, T(n,r)*x^n) and composition G(A(x)) = g(0)+sum(n>0, sum(r=1..n, T(n,r)*g(r))*x^n).

Links

Table of n, a(n) for n=1..66.

Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Formula

T(n,r) = sum(m=r..n, sum(k=m..n,sum(i=k..n, binomial(i,n-i)*binomial(i-1,k-1)) *sum(j=0..m, binomial(m,j) *binomial(j,k-3*m+2*j))) *binomial(r,m-r)), n>0, 1<=r<=n.

Example

Triangle begins:
1;
4,1;
14,8,1;
46,44,12,1;
141,204,90,16,1;
409,846,538,152,20,1;
1132,3234,2787,1112,230,24,1;
3011,11600,13035,6892,1990,324,28,1;

Prog

(Maxima)
T(n, r):= sum(sum(sum(binomial(i, n-i)*binomial(i-1, k-1), i, k, n) *sum(binomial(m, j) *binomial(j, k-3*m+2*j), j, 0, m), k, m, n) *binomial(r, m-r), m, r, n);

Crossrefs

Sequence in context: A050156 A191584 A231185 * A257501 A096644 A145829

Adjacent sequences: A187052 A187053 A187054 * A187056 A187057 A187058

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Mar 02 2011

Status

approved