0,4

The hyperbinomial transform of a sequence {b} is defined to be the sequence {d} given by d(n) = sum(k=0..n, T(n,k)*b(k)), where T(n,k) = (n-k+1)^(n-k-1)*C(n,k).

Given a table in which the n-th row is the n-th binomial transform of the first row, then the hyperbinomial transform of any diagonal results in the next lower diagonal in the table.

The simplest example of a table of iterated binomial transforms is A009998, with a main diagonal of {1,2,9,64,625,...}; and the hyperbinomial transform of this diagonal gives the next lower diagonal, {1,3,16,125,1296,...}, since 1=(1)*1, 3=(1)*1+(1)*2, 16=(3)*1+(2)*2+(1)*9, 125=(16)*1+(9)*2+(3)*9+(1)*64, etc.

Another example: the hyperbinomial transform maps A065440 into A055541, since HYPERBINOMIAL([1,1,1,8,81,1024,15625]) = [1,2,6,36,320,3750,54432] where e.g.f.: A065440(x)+x = x-x/( LambertW(-x)*(1+LambertW(-x)) ), e.g.f.: A055541(x) = x-x*LambertW(-x).

The m-th iteration of the hyperbinomial transform is given by the triangle of coefficients defined by T_m(n,k) = m*(n-k+m)^(n-k-1)*binomial(n,k).

Example: PARI code for T_m: {a=[1,1,1,8,81,1024,15625]; m=1; b=vector(length(a)); for(n=0,length(a)-1, b[n+1]=sum(k=0,n, m*(n-k+m)^(n-k-1)*binomial(n,k)*a[k+1]); print1(b[n+1],","))} RETURNS b=[1,2,6,36,320,3750,54432].

The INVERSE hyperbinomial transform is thus given by m=-1: {a=[1,2,6,36,320,3750,54432]; m=-1; b=vector(length(a)); for(n=0,length(a)-1, b[n+1]=sum(k=0,n, m*(n-k+m)^(n-k-1)*binomial(n,k)*a[k+1]); print1(b[n+1],","))} RETURNS b=[1,1,1,8,81,1024,15625].

Simply stated, the HYPERBINOMIAL transform is to -LambertW(-x)/x as the BINOMIAL transform is to exp(x).

T. D. Noe, Rows n=0..50 of triangle, flattened

T(n, k) = (n-k+1)^(n-k-1)*C(n, k).

E.g.f.: -LambertW(-x)*exp(x*y)/x. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 27 2003

Rows begin:

{1},

{1,1},

{3,2,1},

{16,9,3,1},

{125,64,18,4,1},

{1296,625,160,30,5,1},

{16807,7776,1875,320,45,6,1},

{262144,117649,27216,4375,560,63,7,1},..

Cf. A088957 (row sums), A000272 (first column), A009998.

Adjacent sequences: A088953 A088954 A088955 this_sequence A088957 A088958 A088959

Sequence in context: A135876 A136217 A136220 this_sequence A106208 A129377 A136733

nonn,tabl,nice

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 26 2003