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A185139
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Triangle T(n,k) = Sum_{i=1..n} 2^(i-1)*C(n+2*k-i-1, k-1), 1 <= k <= n.
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1
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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
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OFFSET
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1,2
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COMMENTS
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The first term of the m-th row is 2^m-1.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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,
...
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MATHEMATICA
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Table[Sum[2^(j - 1)*Binomial[n + 2*k - j - 1, k - 1], {j, 1, n}], {n,
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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