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A084757
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For n, k > 0, let T(n, k) be given by T(n, 1) = n and T(n, k+1) = k*T(n, k) + 1. a(n) is the sum of the n-th antidiagonal.
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2
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1, 4, 11, 31, 106, 466, 2577, 17151, 132666, 1165310, 11438525, 123981551, 1469997610, 18919751410, 262644893329, 3911200633719, 62186842823250, 1051369907752254, 18832837831656989, 356278889320409303
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OFFSET
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1,2
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LINKS
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FORMULA
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E.g.f.: exp(x)*(exp(-1)*(Ei(1) - Ei(1-x))*x + 1 - log(1-x) + 1/(1-x)) - 1. - Vladeta Jovovic, Jan 06 2005
a(n) = Sum_{=1..n} (A000522(n-1) + (n-1)!*(k-1)).
a(n) = Sum_{k=0..n-1} floor(e*k!) + n*A003422(n) - A003422(n+1). (End)
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EXAMPLE
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1, 2, 5, 16, 65, 326, 1957, ...
2, 3, 7, 22, 89, 446, 2677, ...
3, 4, 9, 28, 113, 566, 3397, ...
4, 5, 11, 34, 137, 686, 4117, ...
...
The antidiagonal rows and sums are:
1 : 1;
2, 2 : 4;
5, 3, 3 : 11;
16, 7, 4, 4 : 31;
65, 22, 9, 5, 5 : 106;
326, 89, 28, 11, 6, 6 : 466;
...
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MATHEMATICA
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A084756[n_, k_]:= Floor[E*(n-1)!] + (k-1)*(n-1)!;
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PROG
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(Magma)
A084756:= func< n, k | Floor(Exp(1)*Factorial(n-1)) + (k-1)*Factorial(n-1) >;
(SageMath)
def A084756(n, k): return floor(e*factorial(n-1)) + (k-1)*factorial(n-1) - int(n==1)
def A084757(n): return sum( A084756(n-k+1, k) for k in range(1, n+1) )
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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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