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A190782
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Triangle T(n,k), read by rows, of the coefficients of x^k in the expansion of Sum_(m=0..n) binomial(x,m) = (a(k)*x^k)/n!, n >= 0, 0 <= k <= n.
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7
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1, 1, 1, 2, 1, 1, 6, 5, 0, 1, 24, 14, 11, -2, 1, 120, 94, 5, 25, -5, 1, 720, 444, 304, -75, 55, -9, 1, 5040, 3828, 364, 1099, -350, 112, -14, 1, 40320, 25584, 15980, -4340, 3969, -1064, 210, -20, 1
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OFFSET
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0,4
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COMMENTS
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There is a strong relation between this triangle and triangle A048994 which deals with the binomial (x,n), this triangle being dealing with the summation of this binomial.
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LINKS
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FORMULA
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T(n,k) = T(n-1,k)+ T(n-1,k-1)- T(n-2,k-1)*(n-1)+ T(n-2,k)*(n-1)^2, T(n,n)=1, T(n,0)= n! for n >= 0.
T(n,k) = T(n-1,k)*n + (A048994(n,k)), T(n,n)= 1, T(n,0)= n! for n>= 0.
E.g.f. of column k: (log(1 + x))^k/(k! * (1 - x)). - Seiichi Manyama, Sep 26 2021
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EXAMPLE
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Triangle begins:
n\k 0 1 2 3 4 5 6 7 8
0 1
1 1 1
2 2 1 1
3 6 5 0 1
4 24 14 11 -2 1
5 120 94 5 25 -5 1
6 720 444 304 -75 55 -9 1
7 5040 3828 364 1099 -350 112 -14 1
8 40320 25584 15980 -4340 3969 -1064 210 -20 1
...
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MATHEMATICA
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row[n_] := CoefficientList[ Series[ Sum[ Binomial[x, m], {m, 0, n}], {x, 0, n}], x]*n!; Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jan 04 2013 *)
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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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