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A076571
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Binomial triangle based on factorials.
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6
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1, 1, 2, 2, 3, 5, 6, 8, 11, 16, 24, 30, 38, 49, 65, 120, 144, 174, 212, 261, 326, 720, 840, 984, 1158, 1370, 1631, 1957, 5040, 5760, 6600, 7584, 8742, 10112, 11743, 13700, 40320, 45360, 51120, 57720, 65304, 74046, 84158, 95901, 109601
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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LINKS
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FORMULA
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T(n, k) = Sum_{j=0..k} binomial(k, j)*(n-j)!.
T(n, k) = T(n, k-1) + T(n-1, k-1) with T(n, 0) = n!.
Sum_{k=0..n} T(n, k) = A002627(n+1).
T(n, k) = n! * Hypergeometric1F1([-k], [-n], 1).
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EXAMPLE
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Rows start:
1;
1, 2;
2, 3, 5;
6, 8, 11, 16;
24, 30, 38, 49, 65;
120, 144, 174, 212, 261, 326;
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MATHEMATICA
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A076571[n_, k_]:= n!*Hypergeometric1F1[-k, -n, 1];
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PROG
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(Magma)
A076571:= func< n, k| (&+[Binomial(k, j)*Factorial(n-j): j in [0..k]]) >;
(SageMath)
def A076571(n, k): return sum(binomial(k, j)*factorial(n-j) for j in range(k+1))
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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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