login
A185139
Triangle T(n,k) = Sum_{i=1..n} 2^(i-1)*C(n+2*k-i-1, k-1), 1 <= k <= n.
1
1, 3, 10, 7, 25, 91, 15, 56, 210, 792, 31, 119, 456, 1749, 6721, 63, 246, 957, 3718, 14443, 56134, 127, 501, 1969, 7722, 30251, 118456, 463828, 255, 1012, 4004, 15808, 62322, 245480, 966416, 3803648, 511, 2035, 8086, 32071, 127024, 502588, 1987096, 7852453, 31020445, 1023, 4082, 16263
OFFSET
1,2
COMMENTS
The first term of the m-th row is 2^m-1.
LINKS
FORMULA
2*T_n(k) = T_(n-1)(k+1) + C(n+2*k-1,k).
T_n(k) = T_(n-2)(k+1) + C(n+2*k-1,k).
T_n(k) = 2*T_(n-1)(k) + C(n+2*k-2,k-1).
T_n(k+1) = 4*T_n(k) - (n/k)*C(n+2*k-1,k-1).
EXAMPLE
Triangle begins
1,
3, 10,
7, 25, 91,
15, 56, 210, 792,
31, 119, 456, 1749, 6721,
63, 246, 957, 3718, 14443, 56134,
127, 501, 1969, 7722, 30251, 118456, 463828,
255, 1012, 4004, 15808, 62322, 245480, 966416, 3803648,
511, 2035, 8086, 32071, 127024, 502588, 1987096, 7852453, 31020445,
...
MATHEMATICA
Table[Sum[2^(j - 1)*Binomial[n + 2*k - j - 1, k - 1], {j, 1, n}], {n,
1, 50}, {k, 1, n}] // Flatten (* G. C. Greubel, Jun 23 2017 *)
PROG
(PARI) for(n=1, 20, for(k=1, n, print1(sum(j=1, n, 2^(j-1)*binomial(n+2*k-j-1, k-1)), ", "))) \\ G. C. Greubel, Jun 23 2017
CROSSREFS
Cf. A174531.
Sequence in context: A261836 A301937 A373866 * A300786 A182241 A281178
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved