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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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%I #54 May 27 2024 09:18:44

%S 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,

%T 55,-9,1,5040,3828,364,1099,-350,112,-14,1,40320,25584,15980,-4340,

%U 3969,-1064,210,-20,1

%N 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.

%C 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.

%C Apparently A054651 with reversed rows. - _Mathew Englander_, May 17 2014

%H Seiichi Manyama, <a href="/A190782/b190782.txt">Rows n = 0..139, flattened</a>

%F 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.

%F T(n,k) = T(n-1,k)*n + (A048994(n,k)), T(n,n)= 1, T(n,0)= n! for n>= 0.

%F E.g.f. of column k: (log(1 + x))^k/(k! * (1 - x)). - _Seiichi Manyama_, Sep 26 2021

%F T(n, k) = Sum_{i=0..n-k} Stirling1(i+k, k)*n!/(i+k)!. - _Igor Victorovich Statsenko_, May 27 2024

%e Triangle begins:

%e n\k 0 1 2 3 4 5 6 7 8

%e 0 1

%e 1 1 1

%e 2 2 1 1

%e 3 6 5 0 1

%e 4 24 14 11 -2 1

%e 5 120 94 5 25 -5 1

%e 6 720 444 304 -75 55 -9 1

%e 7 5040 3828 364 1099 -350 112 -14 1

%e 8 40320 25584 15980 -4340 3969 -1064 210 -20 1

%e ...

%t 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 *)

%Y T(2*n,n) gives A347987.

%Y Column 0-5 give A000142, A024167, A348063, A348064, A348065, A348068.

%Y Cf. A048994, A054651, A132393.

%K sign,tabl

%O 0,4

%A _Mokhtar Mohamed_, Dec 29 2012