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A276659
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Accumulation of the upper left triangle used in binomial transform of nonnegative integers.
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1
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0, 2, 11, 39, 114, 300, 741, 1757, 4052, 9162, 20415, 44979, 98214, 212888, 458633, 982905, 2097000, 4456278, 9436995, 19922735, 41942810, 88080132, 184549101, 385875669, 805306044, 1677721250, 3489660551, 7247756907, 15032385102, 31138512432, 64424508945
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OFFSET
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0,2
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COMMENTS
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After 0, is this the second column of A108284? [Bruno Berselli, Sep 13 2016 - this comment may be removed if the property is confirmed.]
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LINKS
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FORMULA
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O.g.f.: x*(2 - 3*x)/((1 - x)^3*(1 - 2*x)^2).
E.g.f.: x*exp(x)*(8*exp(x) - x - 4)/2.
a(n) = n*(2^(n+2) - n - 3)/2.
a(n) = 7*a(n-1) - 19*a(n-2) + 25*a(n-3) - 16*a(n-4) + 4*a(n-5) for n > 4.
a(n) = Sum_{k=2..n+3} Sum_{i=2..n+3} k * C(n-i+3,k). - Wesley Ivan Hurt, Sep 20 2017
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EXAMPLE
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Starting from the triangle:
0, 1, 2, 3, 4, 5, ...
1, 3, 5, 7, 9, ...
4, 8, 12, 16, ...
12, 20, 28, ...
32, 48, ...
80, ...
...
the first terms are:
a(0) = 0;
a(1) = a(0) + 1 + 1 = 2;
a(2) = a(1) + 4 + 3 + 2 = 11;
a(3) = a(2) + 12 + 8 + 5 + 3 = 39, etc.
First column is A001787: n*2^(n-1).
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MAPLE
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MATHEMATICA
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t[0, k_] := k; t[n_, k_] := t[n, k] = t[n - 1, k] + t[n - 1, k + 1]; a[n_] := Sum[t[m, k], {m, 0, n}, {k, 0, n - m}]; Table[a[n], {n, 0, 30}]
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PROG
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(PARI) x='x+O('x^99); concat(0, Vec(x*(2-3*x)/((1-x)^3*(1-2*x)^2))) \\ Altug Alkan, Sep 14 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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