# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a238394 Showing 1-1 of 1 %I A238394 #29 Oct 27 2023 20:51:06 %S A238394 1,0,1,1,1,2,3,3,4,5,8,9,12,13,17,22,28,34,42,48,59,71,88,106,130,151, %T A238394 181,210,250,295,354,417,494,577,675,780,909,1053,1231,1431,1668,1930, %U A238394 2240,2573,2963,3392,3896,4461,5129,5873,6742,7710,8816,10043,11439 %N A238394 Number of partitions of n that sorted in increasing order do not contain a part k in position k. %C A238394 The definition forbids partitions with a part equal to 1, so the smallest possible part is 2, which however can appear at most once. %C A238394 Note that considering partitions in standard decreasing order, we obtain A064428. %H A238394 Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Giovanni Resta) %F A238394 a(n) + A238395(n) = p(n) = A000041(n). %F A238394 a(n) = Sum_{k>=0} A238406(n,k). - _Alois P. Heinz_, Feb 26 2014 %F A238394 a(n) = A238352(n,0). - _Alois P. Heinz_, Jun 08 2014 %e A238394 a(6) = 3, because of the 11 partitions of 6 only 3 do not contain a 1 in position 1, a 2 in position 2, or a 3 in position 3, namely (3,3), (2,4) and (6). %e A238394 From _Joerg Arndt_, Mar 23 2014: (Start) %e A238394 There are a(15) = 22 such partitions of 15: %e A238394 01: [ 2 3 4 6 ] %e A238394 02: [ 2 3 5 5 ] %e A238394 03: [ 2 3 10 ] %e A238394 04: [ 2 4 4 5 ] %e A238394 05: [ 2 4 9 ] %e A238394 06: [ 2 5 8 ] %e A238394 07: [ 2 6 7 ] %e A238394 08: [ 2 13 ] %e A238394 09: [ 3 3 4 5 ] %e A238394 10: [ 3 3 9 ] %e A238394 11: [ 3 4 8 ] %e A238394 12: [ 3 5 7 ] %e A238394 13: [ 3 6 6 ] %e A238394 14: [ 3 12 ] %e A238394 15: [ 4 4 7 ] %e A238394 16: [ 4 5 6 ] %e A238394 17: [ 4 11 ] %e A238394 18: [ 5 5 5 ] %e A238394 19: [ 5 10 ] %e A238394 20: [ 6 9 ] %e A238394 21: [ 7 8 ] %e A238394 22: [ 15 ] %e A238394 (End) %p A238394 b:= proc(n, i) option remember; `if`(n=0, 1, %p A238394 `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, (p-> expand( %p A238394 x*(p-coeff(p, x, i-1)*x^(i-1))))(b(n-i, i))))) %p A238394 end: %p A238394 a:= n-> (p-> add(coeff(p, x, i), i=0..degree(p)))(b(n$2)): %p A238394 seq(a(n), n=0..70); # _Alois P. Heinz_, Feb 26 2014 %t A238394 a[n_] := Length@ Select[ IntegerPartitions@n, 0 < Min@ Abs[ Reverse@# - Range@ Length@#] &]; Array[a, 30] %t A238394 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, Expand[x*(p-Coefficient[p, x, i-1]*x^(i-1))]][b[n-i, i]]]]]; a[n_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Exponent[p, x]} ] ][b[n, n]]; Table[a[n], {n, 0, 70}] (* _Jean-François Alcover_, Nov 02 2015, after _Alois P. Heinz_ *) %Y A238394 Cf. A000041, A238395, A064428, A001522, A238352. %K A238394 nonn %O A238394 0,6 %A A238394 _Giovanni Resta_, Feb 26 2014 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE