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 A134400 #55 Dec 03 2023 17:18:23
%S A134400 1,1,1,2,4,2,3,9,9,3,4,16,24,16,4,5,25,50,50,25,5,6,36,90,120,90,36,6,
%T A134400 7,49,147,245,245,147,49,7,8,64,224,448,560,448,224,64,8,9,81,324,756,
%U A134400 1134,1134,756,324,81,9,10,100,450,1200,2100,2520,2100,1200,450,100,10
%N A134400 M * A007318, where M = triangle with (1, 1, 2, 3, ...) in the main diagonal and the rest zeros.
%C A134400 Row sums = A134401: (1, 2, 8, 24, 64, 160, 384, ...).
%C A134400 Triangle T(n,k), read by rows, given by [1,1,-1,1,0,0,0,0,0,...] DELTA [1,1,-1,1,0,0,0,0,0,...] where DELTA is the operator defined in A084938. A134402*A007318 as infinite lower triangular matrices. - _Philippe Deléham_, Oct 26 2007
%C A134400 For n > 0, from n athletes, select a team of k players and then choose a coach who is allowed to be on the team or not. - _Geoffrey Critzer_, Mar 13 2010
%C A134400 Row sums are A036289 if first term changed to zero. Diagonal sums are A023610, starting with the 2nd diagonal. Partial sums of diagonals are A002940 if first term changed to zero. - _John Molokach_, Jul 06 2013
%C A134400 For n > 0, T(n,k) is the number of states in Sokoban puzzle with n non-obstacles cells and k boxes (see Russell and Norvig at page 157). - _Stefano Spezia_, Dec 03 2023
%D A134400 Stuart Russell and Peter Norvig, Artificial Intelligence: A Modern Approach, Fourth Edition, Hoboken: Pearson, 2021.
%H A134400 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sokoban">Sokoban</a>
%F A134400 From _Geoffrey Critzer_, Mar 13 2010: (Start)
%F A134400 T(0,0) = 1 and T(n,k) = n * binomial(n,k) for n > 0.
%F A134400 E.g.f. for column k is: (x^k/k!)*exp(x)*(x+k). (End)
%F A134400 T(n,k) = A003506(n,k) + A003506(n,k-1). - _Geoffrey Critzer_, Mar 13 2010
%F A134400 G.f.: (1-x-x*y+x^2+x^2*y+x^2*y^2)/(1-2*x-2*x*y+x^2+2*x^2*y+x^2*y^2). - _Philippe Deléham_, Nov 14 2013
%F A134400 T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) - T(n-2,k-2), T(0,0)=T(1,0)=T(1,1)=1, T(2,0)=T(2,2)=2, T(2,1)=4, T(n,k)=0 if k < 0 or if k > n. - _Philippe Deléham_, Nov 14 2013
%F A134400 E.g.f.: 1 + exp(y*x)*exp(x)*(y*x + x). - _Geoffrey Critzer_, Mar 15 2015
%e A134400 First few rows of the triangle:
%e A134400 1;
%e A134400 1, 1;
%e A134400 2, 4, 2;
%e A134400 3, 9, 9, 3;
%e A134400 4, 16, 24, 16, 4;
%e A134400 5, 25, 50, 50, 25, 5;
%e A134400 6, 36, 90, 120, 90, 36, 6;
%e A134400 7, 49, 147, 245, 245, 147, 49, 7;
%e A134400 ...
%p A134400 with(combstruct): for n from 0 to 10 do seq(`if`(n=0, 1, n)* count( Combination(n), size=m), m=0..n) od; # _Zerinvary Lajos_, Apr 09 2008
%t A134400 Join[{1},Table[Table[n*Binomial[n, k], {k,0, n}], {n, 10}]] //Flatten (* _Geoffrey Critzer_, Mar 13 2010 adapted by _Stefano Spezia_, Dec 03 2023 *)
%Y A134400 Cf. A002940, A003506, A007318, A036289, A134401, A134402.
%Y A134400 T(2n,n) give A002011(n-1) for n>=1.
%K A134400 nonn,tabl
%O A134400 0,4
%A A134400 _Gary W. Adamson_, Oct 23 2007
%E A134400 a(55)-a(65) from _Stefano Spezia_, Dec 03 2023