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A088956
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Triangle, read by rows, of coefficients of the hyperbinomial transform.
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16
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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, 4782969, 2097152, 470596, 72576, 8750, 896, 84, 8, 1, 100000000, 43046721, 9437184, 1411788, 163296, 15750
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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).
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LINKS
| T. D. Noe, Rows n=0..50 of triangle, flattened
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FORMULA
| 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.rs), Oct 27 2003
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EXAMPLE
| 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},..
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CROSSREFS
| Cf. A088957 (row sums), A000272 (first column), A009998.
Sequence in context: A136217 A166884 A136220 * A106208 A129377 A136733
Adjacent sequences: A088953 A088954 A088955 * A088957 A088958 A088959
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KEYWORD
| nonn,tabl,nice
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Oct 26 2003
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