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 A102639 #11 Feb 07 2019 17:24:43
%S A102639 1,1,3,8,8,3,9,46,101,114,65,15,33,272,975,1935,2289,1615,630,105,153,
%T A102639 1796,9175,26795,49474,60080,48104,24535,7245,945,873,13424,90255,
%U A102639 353507,902164,1582455,1953272,1700860,1025927,408870,97020,10395,5913
%N A102639 Combinatorial triangle !n. This table read by rows gives the coefficients of general sum formulas of n-th left factorials (A003422). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k and k=1 to n-2, where T(i,k) satisfies !n = n + Sum_{k=1..n-2} Sum_{i=1..2*k} T(i,k) * C(n-k-1,i).
%C A102639 The coefficients T(i,k) along the i-th columns of the triangle are the consecutive partial sums of those found in table A094216.
%H A102639 Chris Zheng, Jeffrey Zheng, <a href="https://doi.org/10.1007/978-981-13-2282-2_4">Triangular Numbers and Their Inherent Properties</a>, Variant Construction from Theoretical Foundation to Applications, Springer, Singapore, 51-65.
%e A102639 !7 = 7 + 1*C(7-2,1) + 1*C(7-2,2) + 3*C(7-3,1) + ... + 33*C(7-5,1) + 272*C(7-5,2) + 153*C(7-6,1) = 7 + 5 + 10 + 12 + 8*C(4,2) + 8*C(4,3) + 3*C(4,4) + 9*C(3,1) + 46*C(3,2) + 101*C(3,3) + 66 + 272 + 153 = 7 + 5 + 10 + 12 + 48 + 32 + 3 + 27 + 138 + 101 + 66 + 272 + 153 = 874.
%Y A102639 Cf. A094216, A102411, A102412, A101752, A003422, A094638, A008276.
%K A102639 nonn,tabl
%O A102639 1,3
%A A102639 _André F. Labossière_, Feb 01 2005