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 A117670 #42 Nov 25 2015 20:04:35
%S A117670 1,2,3,3,6,7,4,10,14,15,5,15,25,30,31,6,21,41,56,62,63,7,28,63,98,119,
%T A117670 126,127,8,36,92,162,218,246,254,255,9,45,129,255,381,465,501,510,511,
%U A117670 10,55,175,385,637,847,967,1012,1022,1023
%N A117670 Triangle read by rows: partial sums of the Pascal triangle minus 1.
%C A117670 Imagine that you are in a building with floors starting at floor 1, the lowest floor and you have a large number of eggs. For each floor in the building, you want to know whether or not an egg dropped from that floor will break.
%C A117670 If an egg breaks when dropped from floor i, then all eggs are guaranteed to break when dropped from any floor j > i. Likewise, if an egg doesn't break when dropped from floor i, then all eggs are guaranteed to never break when dropped from any floor j <= i.
%C A117670 a(n,k) is the maximum number of floors where you can determine whether or not an egg will break when dropped from any floor, with the following restrictions: you may drop a maximum of n eggs (one at a time, from any floors of your choosing) and you may break a maximum of k eggs.
%C A117670 Each row of the triangle is the running sum of the corresponding row with the first 1 omitted of Pascal's triangle (A007318), see A008949, A054143, A193820.
%C A117670 The k-th entry in the n-th row is the number of possible combinations of on/off switches after k attempts to turn on a switch in a set of n distinguishable switches. An attempt to turn on the same switch twice does not result in a new combination. See example. - _Sergei Viznyuk_, Jun 24 2012
%C A117670 T(n,k) is the number of nonempty subsets of the n-set with at most k elements, see example. - _Joerg Arndt_, May 04 2014
%H A117670 Susanne Wienand, <a href="/A117670/b117670.txt">Table of n, a(n) for n = 1..1830</a>
%H A117670 Google code.jam, <a href="https://code.google.com/codejam/contest/dashboard?c=agdjb2RlamFtcg4LEghjb250ZXN0cxh5DA#s=p2">Problem C. Egg Drop</a>
%H A117670 Sergei Viznyuk, <a href="http://phystech.com/ftp/A117670.c">C-Program</a>
%H A117670 Sergei Viznyuk, <a href="/A117670/a117670.c.txt">Local copy of C-Program</a>
%F A117670 a(n,1) = n ; a(n,n) = 2^n-1; a(n+1,k+1) = 1 + a(n,k) + a(n,k-1), 0 < k < n.
%F A117670 a(n,k) = sum(binomial(n,m),m=1..k), 1 <= k <= n. (see the running sum comment above). - _Wolfdieter Lang_, Feb 07 2013
%e A117670 Triangle a(n,k) begins:
%e A117670 n\k 1 2 3 4 5 6 7 8 9 10 ...
%e A117670 1: 1
%e A117670 2: 2 3
%e A117670 3: 3 6 7
%e A117670 4: 4 10 14 15
%e A117670 5: 5 15 25 30 31
%e A117670 6: 6 21 41 56 62 63
%e A117670 7: 7 28 63 98 119 126 127
%e A117670 8: 8 36 92 162 218 246 254 255
%e A117670 9: 9 45 129 255 381 465 501 510 511
%e A117670 10: 10 55 175 385 637 847 967 1012 1022 1023
%e A117670 ... Reformatted and extended by _Wolfdieter Lang_, Feb 07 2013
%e A117670 From _Sergei Viznyuk_, Jun 24 2012: (Start)
%e A117670 For example, we have n=3 distinguishable switches A,B,C (third row above). We attempt k=2 times to turn on a switch at random. The possible resulting combinations are:
%e A117670 A=on, B=off, C=off (the same A switch was turned on 2 times)
%e A117670 A=off, B=on, C=off (the same B switch was turned on 2 times)
%e A117670 A=off, B=off, C=on (the same C switch was turned on 2 times)
%e A117670 A=on, B=on, C=off (switches A and B were turned on)
%e A117670 A=on, B=off, C=on (switches A and C were turned on)
%e A117670 A=off, B=on, C=on (switches B and C were turned on)
%e A117670 Thus, we have 6 different combinations, which is the number 6 at row n=3 column k=2 in the sequence above.
%e A117670 (End)
%e A117670 From _Joerg Arndt_, May 04 2014: (Start)
%e A117670 There are T(4,2) = 10 subsets of {0, 1, 2, 3}:
%e A117670 01: 1... { 0 }
%e A117670 02: 11.. { 0, 1 }
%e A117670 03: 111. { 0, 1, 2 }
%e A117670 04: 11.1 { 0, 1, 3 }
%e A117670 05: 1.1. { 0, 2 }
%e A117670 06: 1.11 { 0, 2, 3 }
%e A117670 07: 1..1 { 0, 3 }
%e A117670 08: .1.. { 1 }
%e A117670 09: .11. { 1, 2 }
%e A117670 10: .111 { 1, 2, 3 }
%e A117670 11: .1.1 { 1, 3 }
%e A117670 12: ..1. { 2 }
%e A117670 13: ..11 { 2, 3 }
%e A117670 14: ...1 { 3 }
%e A117670 (End)
%t A117670 Table[Sum[Binomial[n, m], {m, k}], {n, 10}, {k, n}] // Flatten (* _Michael De Vlieger_, Nov 25 2015 *)
%o A117670 (PARI) tabl(nrows) = {for (n=1, nrows, for (k=1, n, print1(sum(m=1,k,binomial(n,m)), ", ");); print(););} \\ _Michel Marcus_, May 21 2013
%K A117670 nonn,tabl
%O A117670 1,2
%A A117670 _Arie Bos_, Jul 06 2008, Jul 08 2008