This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A190782 #54 May 27 2024 09:18:44
%S A190782 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 A190782 55,-9,1,5040,3828,364,1099,-350,112,-14,1,40320,25584,15980,-4340,
%U A190782 3969,-1064,210,-20,1
%N A190782 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 A190782 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 A190782 Apparently A054651 with reversed rows. - _Mathew Englander_, May 17 2014
%H A190782 Seiichi Manyama, <a href="/A190782/b190782.txt">Rows n = 0..139, flattened</a>
%F A190782 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 A190782 T(n,k) = T(n-1,k)*n + (A048994(n,k)), T(n,n)= 1, T(n,0)= n! for n>= 0.
%F A190782 E.g.f. of column k: (log(1 + x))^k/(k! * (1 - x)). - _Seiichi Manyama_, Sep 26 2021
%F A190782 T(n, k) = Sum_{i=0..n-k} Stirling1(i+k, k)*n!/(i+k)!. - _Igor Victorovich Statsenko_, May 27 2024
%e A190782 Triangle begins:
%e A190782 n\k 0 1 2 3 4 5 6 7 8
%e A190782 0 1
%e A190782 1 1 1
%e A190782 2 2 1 1
%e A190782 3 6 5 0 1
%e A190782 4 24 14 11 -2 1
%e A190782 5 120 94 5 25 -5 1
%e A190782 6 720 444 304 -75 55 -9 1
%e A190782 7 5040 3828 364 1099 -350 112 -14 1
%e A190782 8 40320 25584 15980 -4340 3969 -1064 210 -20 1
%e A190782 ...
%t A190782 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 A190782 T(2*n,n) gives A347987.
%Y A190782 Column 0-5 give A000142, A024167, A348063, A348064, A348065, A348068.
%Y A190782 Cf. A048994, A054651, A132393.
%K A190782 sign,tabl
%O A190782 0,4
%A A190782 _Mokhtar Mohamed_, Dec 29 2012