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Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.
T(n, k) = (k+n+1)*(n+1)^(k-1). E.g.f.: (1+x)*exp(x)/((1-y*exp(x)).
E.g.f.: (1+x)*exp(x)/((1-y*exp(x)).
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editing
editing
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(PARI) T(n, k)=if(n<0 || k<0, 0, (k+n+1)*(n+1)^(k-1))
approved
editing
_Paul D. Hanna (pauldhanna(AT)juno.com), _, Nov 23 2003
nonn,tabl,new
Paul D . Hanna (pauldhanna(AT)juno.com), Nov 23 2003
T(n, k)=(k+n+1)*(n+1)^(k-1). E.g.f.: (1+x)*exp(x)/((1-y*exp(x)).
nonn,tabl,new
Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k)=(k+1) for k>=0.
1, 2, 1, 3, 3, 1, 4, 8, 4, 1, 5, 20, 15, 5, 1, 6, 48, 54, 24, 6, 1, 7, 112, 189, 112, 35, 7, 1, 8, 256, 648, 512, 200, 48, 8, 1, 9, 576, 2187, 2304, 1125, 324, 63, 9, 1, 10, 1280, 7290, 10240, 6250, 2160, 490, 80, 10, 1, 11, 2816, 24057, 45056, 34375, 14256, 3773, 704
0,2
T(n,k)=(k+n+1)*(n+1)^(k-1). E.g.f.: (1+x)*exp(x)/((1-y*exp(x)).
Rows begin:
{1, 2, 3, 4, 5, 6, 7,..},
{1, 3, 8, 20, 48, 112, 256,..},
{1, 4, 15, 54, 189, 648, 2187,..},
{1, 5, 24, 112, 512, 2304, 10240,..},
{1, 6, 35, 200, 1125, 6250, 34375,..},
{1, 7, 48, 324, 2160, 14256, 93312,..},
{1, 8, 63, 490, 3773, 28812, 218491,..},..
(PARI) T(n, k)=if(n<0|k<0, 0, (k+n+1)*(n+1)^(k-1))
nonn,tabl
Paul D Hanna (pauldhanna(AT)juno.com), Nov 23 2003
approved