A202039 Triangle T(n,m) = coefficient of x^n in expansion of (1/2-1/2*(1-8*x)^1/4)^m = sum(n>=m, T(n,m) x^n), n>=1, m>=1.

1, 3, 1, 14, 6, 1, 77, 37, 9, 1, 462, 238, 69, 12, 1, 2926, 1582, 510, 110, 15, 1, 19228, 10780, 3738, 920, 160, 18, 1, 129789, 74877, 27405, 7389, 1495, 219, 21, 1, 894102, 528022, 201569, 58156, 13075, 2262, 287, 24, 1, 6258714, 3769370, 1488762, 452826, 110143, 21417, 3248, 364, 27, 1

Offset

1,2

Links

Table of n, a(n) for n=1..55.

Lili Mu, Yuanyuan Xing, and Sai-Nan Zheng, A New Criterion for the Total Positivity of Riordan Arrays, Journal of Integer Sequences, Vol. 28 (2025), Article 25.7.5. See p. 8.

Formula

T(n,m) = (m*Sum_{k=0..n-m} (-1)^(n-m-k)*binomial(n+k-1,n-1)*Sum_{j=0..k} binomial(j,n-m+(-3)*k+2*j)*binomial(k,j)*2^(2*n-2*m+(-5)*k+3*j)*3^(-n+m+3*k-j))/n.

T(n,m) = (m*Sum_{k=m..n} binomial(-m+2*k-1,k-1)*2^(n-k)*binomial(2*n-k-1,n-1))/n. - Vladimir Kruchinin, Dec 21 2011

T(n,m) = (m/n)*2^(n-m)*binomial(2*n-m-1,n-m)*hypergeometric([1/2+m/2,m/2,m-n],[m,1+m-2*n],2) for n>1, m>1. - Peter Luschny, Jan 04 2012

Example

1,
3, 1,
14, 6, 1,
77, 37, 9, 1,
462, 238, 69, 12, 1,
2926, 1582, 510, 110, 15, 1

Prog

(Maxima)
T(n, m):=(m*sum((-1)^(n-m-k)*binomial(n+k-1, n-1)*sum(binomial(j, n-m+(-3)*k+2*j)*binomial(k, j)*2^(2*n-2*m+(-5)*k+3*j)*3^(-n+m+3*k-j), j, 0, k), k, 0, n-m))/n;
T(n, m):=(m*sum(binomial(-m+2*k-1, k-1)*2^(n-k)*binomial(2*n-k-1, n-1), k, m, n))/n;

Crossrefs

Sequence in context: A164807 A283285 A283374 * A122689 A283402 A283349

Adjacent sequences: A202036 A202037 A202038 * A202040 A202041 A202042

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Dec 10 2011

Status

approved