OFFSET
1,2
COMMENTS
The first term of the m-th row is 2^m-1.
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
V. Shevelev and P. Moses, On a sequence of polynomials with hypothetically integer coefficients arXiv:1112.5715 [math.NT], 2011.
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
KEYWORD
nonn,tabl
AUTHOR
Vladimir Shevelev and Peter J. C. Moses, Feb 04 2012
STATUS
approved