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 A078751 #11 Jul 14 2025 20:49:09
%S A078751 2,4,8,24,48,48,224,480,576,384,2880,6400,8640,7680,3840,47232,107520,
%T A078751 155520,161280,115200,46080,942592,2182656,3306240,3763200,3225600,
%U A078751 1935360,645120,22171648,51996672,81414144,98703360,94617600,69672960
%N A078751 Triangle read by rows: T(m,k) = normalized partial derivative of (t,z) -> exp(t*g(z)) at (0,0), where 2*g(z) = 1 + exp(-2*z*g(z)).
%C A078751 Let g(z) = 1/2 + W(z/e^z) / (2*z), where W is Lambert's W-function; g satisfies 2 * g(z) = 1 + exp(-2 * z *(z)). Let c(m,n) be the coefficient of z^m in the Maclaurin series for g(z)^n; equivalently c(m,n) is 1/m! times the mixed partial derivative (d^(m+n) f(t,z)) / (dz^m dt^n), where f(t,z) = exp(t*g(z)). For 0<k<=m, let T(m,k) = 2^k * m! * (-1)^(m-k) * c(m-k,k). The sequence gives the values of T(m, k) read by rows.
%F A078751 T(n, k) = 2^k * n! * (-1)^(n-k) * c(n-k,k) where c(n, k) = (1/n) * Sum_{j=1..n} (((k+1)*j-n) * c(n-j, k) * c(j, 1)), where c(0, k)=1 and c(j, 1) = (1/2) * (-1)^j * (1/(j+1)!) * Sum_{i=1..j+1} binomial(j+1, i) * i^j.
%e A078751 Triangle begins:
%e A078751 2;
%e A078751 4, 8;
%e A078751 24, 48, 48;
%e A078751 224, 480, 576, 384;
%e A078751 ...
%t A078751 (* ccctri lists first numrows rows of triangular array. *)
%t A078751 ccctri[numrows_] := (s[j_] := Sum[Binomial[j, i] i^(j-1), {i, 1, j}]; r[j_] := 1/2 (-1)^j 1/(j+1)! s[j+1]; c[m_, k_] := 1/m Sum[((k+1) j-m)c[m-j, k]r[j], {j, 1, m}]; c[0, k_] := 1; ss[m_, k_] := 2^k m! (-1)^(m-k) c[m-k, k]; Flatten[Table[Table[ss[k, j], {j, 1, k}], {k, 1, numrows}]])
%t A078751 (* ccccol lists maxrow elements of column colnum. *)
%t A078751 ccccol[colnum_, maxrow_] := (s[j_] := Sum[Binomial[j, i] i^(j-1), {i, 1, j}]; r[j_] := 1/2 (-1)^j 1/(j+1)! s[j+1]; c[m_, k_] := 1/m Sum[((k+1) j-m)c[m-j, k]r[j], {j, 1, m}]; c[0, k_] := 1; ss[m_, k_] := 2^k m! (-1)^(m-k) c[m-k, k]; Table[ss[m, colnum], {m, colnum, maxrow}])
%Y A078751 First column of triangular array (T(m, 1) for m>=1) is A038049.
%K A078751 nonn,tabl,easy
%O A078751 0,1
%A A078751 Carmen Chicone (carmen(AT)chicone.math.missouri.edu), Dec 22 2002
%E A078751 Edited by _Dean Hickerson_, Dec 30 2002
%E A078751 Revised by _Sean A. Irvine_, Jul 14 2025