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 A225114 #32 Sep 22 2024 03:08:32
%S A225114 1,1,3,9,28,87,272,850,2659,8318,26025,81427,254777,797175,2494307,
%T A225114 7804529,24419909,76408475,239077739,748060606,2340639096,7323726778,
%U A225114 22915525377,71701378526,224349545236,701976998795,2196446204672,6872555567553,21503836486190,67284284442622,210528708959146
%N A225114 Number of skew partitions of n whose diagrams have no empty rows and columns.
%C A225114 A skew partition S of size n is a pair of partitions [p1,p2] where p1 is a partition of the integer n1, p2 is a partition of the integer n2, p2 is an inner partition of p1, and n=n1-n2. We say that p1 and p2 are respectively the inner and outer partitions of S. A skew partition can be depicted by a diagram made of rows of cells, in the same way as a partition. Only the cells of the outer partition p1 which are not in the inner partition p2 appear in the picture. [from the Sage manual, see links]
%H A225114 Sage Development Team, <a href="http://www.sagemath.org/doc/reference/combinat/sage/combinat/skew_partition.html">Skew Partitions</a>, Sage Reference Manual.
%F A225114 Conjectured g.f.: 1/(2 - 1/(1 - x/(1 - x/(1 - x^2/(1 - x^2/(1 - x^3/(1 - x^3/(1 - ...)))))))). - _Mikhail Kurkov_, Sep 03 2024
%e A225114 The a(4)=28 skew partitions of 4 are
%e A225114 01: [[4], []]
%e A225114 02: [[3, 1], []]
%e A225114 03: [[4, 1], [1]]
%e A225114 04: [[2, 2], []]
%e A225114 05: [[3, 2], [1]]
%e A225114 06: [[4, 2], [2]]
%e A225114 07: [[2, 1, 1], []]
%e A225114 08: [[3, 2, 1], [1, 1]]
%e A225114 09: [[3, 1, 1], [1]]
%e A225114 10: [[4, 2, 1], [2, 1]]
%e A225114 11: [[3, 3], [2]]
%e A225114 12: [[4, 3], [3]]
%e A225114 13: [[2, 2, 1], [1]]
%e A225114 14: [[3, 3, 1], [2, 1]]
%e A225114 15: [[3, 2, 1], [2]]
%e A225114 16: [[4, 3, 1], [3, 1]]
%e A225114 17: [[2, 2, 2], [1, 1]]
%e A225114 18: [[3, 3, 2], [2, 2]]
%e A225114 19: [[3, 2, 2], [2, 1]]
%e A225114 20: [[4, 3, 2], [3, 2]]
%e A225114 21: [[1, 1, 1, 1], []]
%e A225114 22: [[2, 2, 2, 1], [1, 1, 1]]
%e A225114 23: [[2, 2, 1, 1], [1, 1]]
%e A225114 24: [[3, 3, 2, 1], [2, 2, 1]]
%e A225114 25: [[2, 1, 1, 1], [1]]
%e A225114 26: [[3, 2, 2, 1], [2, 1, 1]]
%e A225114 27: [[3, 2, 1, 1], [2, 1]]
%e A225114 28: [[4, 3, 2, 1], [3, 2, 1]]
%o A225114 (Sage) [SkewPartitions(n).cardinality() for n in range(16)]
%o A225114 (PARI) \\ The following program is significantly faster.
%o A225114 A225114(n)=
%o A225114 {
%o A225114 my( C=vector(n, j, 1) );
%o A225114 my(m=n, z, t, ret);
%o A225114 while ( 1, /* for all compositions C[1..m] of n */
%o A225114 \\ print( vector(m, n, C[n] ) ); /* print composition */
%o A225114 t = prod(j=2,m, min(C[j-1], C[j]) + 1 ); /* A225114 */
%o A225114 \\ t = prod(j=2,m, min(C[j-1], C[j]) + 0 ); /* A006958 */
%o A225114 \\ t = prod(j=2,m, C[j-1] + C[j] + 0 ); /* A059716 */
%o A225114 \\ t = prod(j=2,m, C[j-1] + C[j] + 1 ); /* A187077 */
%o A225114 \\ t = sum(j=2,m, C[j-1] > C[j] ); /* A045883 */
%o A225114 ret += t;
%o A225114 if ( m<=1, break() ); /* last composition? */
%o A225114 /* create next composition: */
%o A225114 C[m-1] += 1;
%o A225114 z = C[m];
%o A225114 C[m] = 1;
%o A225114 m += z - 2;
%o A225114 );
%o A225114 return(ret);
%o A225114 }
%o A225114 for (n=0, 30, print1(A225114(n),", "));
%o A225114 \\ _Joerg Arndt_, Jul 09 2013
%K A225114 nonn
%O A225114 0,3
%A A225114 _Joerg Arndt_, Apr 29 2013
%E A225114 Edited by _Max Alekseyev_, Dec 22 2015