cp's OEIS Frontend

A249163 Triangle read by rows: the positive terms of A163626.

%I A249163 #29 Dec 16 2014 13:36:17
%S A249163 1,1,1,2,1,12,1,50,24,1,180,360,1,602,3360,720,1,1932,25200,20160,1,
%T A249163 6050,166824,332640,40320,1,18660,1020600,4233600,1814400,1,57002,
%U A249163 5921520,46070640,46569600,3628800,1,173052,33105600,451725120,898128000,239500800
%N A249163 Triangle read by rows: the positive terms of A163626.
%C A249163 We have two possibilities: with or without 0's.
%C A249163 Without 0's:
%C A249163 1,
%C A249163 1,
%C A249163 1,   2,
%C A249163 1,  12,
%C A249163 1,  50,  24,
%C A249163 1, 180, 360,
%C A249163 etc.
%C A249163 Sum of every row: A000670(n).
%C A249163 First two terms of successive columns: 1, 1, 2, 12, 24, 360, ... = A211374.
%C A249163 With 0's:
%C A249163 1,   0,    0,   0,
%C A249163 1,   0,    0,   0,
%C A249163 1,   2,    0,   0,
%C A249163 1,  12,    0,   0,
%C A249163 1,  50,   24,   0,
%C A249163 1, 180,  360,   0,
%C A249163 1, 602, 3360, 720,
%C A249163 etc.
%C A249163 The columns are essentially A000012, A028243, A028246, A228909, A228911, A228913, from Stirling numbers of the second kind S(n,3), S(n,5), S(n,7), S(n,9), S(n,11), ... .
%t A249163 Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[Derivative[n][y][x], y[x]] // Rest; Table[ Select[row[n], Positive] , {n, 0, 12}] // Flatten
%t A249163 (* or, simply: *) Table[(-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten // Select[#, Positive]& (* _Jean-François Alcover_, Dec 16 2014 *)
%Y A249163 Cf. A163626, A000670, A211374; also A000012, A000392, A000481, A000771, A049447, A028243, A028246, A091137, A228909, A163626, A228911, A228913 and Worpitzky numbers for the second Bernoulli numbers A164555(n)/A027642(n).
%K A249163 nonn,tabf
%O A249163 0,4
%A A249163 _Paul Curtz_, Dec 15 2014