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A078751
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Triangle read by rows: ss(m,k) = normalized partial derivative of (t,z)->exp(t g(z)) at (0,0), where 2 g(z)=1+exp(-2 z g(z)).
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0
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2, 4, 8, 24, 48, 48, 224, 480, 576, 384, 2880, 6400, 8640, 7680, 3840, 47232, 107520, 155520, 161280, 115200, 46080, 942592, 2182656, 3306240, 3763200, 3225600, 1935360, 645120, 22171648, 51996672, 81414144, 98703360, 94617600, 69672960
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OFFSET
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0,1
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COMMENTS
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Let g(z) = 1/2 + W(z/e^z)/(2 z), where W is Lambert's W-function; g satisfies 2 g(z)=1+exp(-2 z g(z)). Let c(m,n) be the coefficient of z^m in the Maclaurin series for g(z)^n; equivalently c(m,n) is 1/m! times the mixed partial derivative (d^(m+n) f(t,z))/(dz^m dt^n), where f(t,z) = exp(t g(z)). For 0<k<=m, let ss(m,k) = 2^k m! (-1)^(m-k) c(m-k,k). The sequence gives the values of ss in the order ss(1,1); ss(2,1), ss(2,2); ss(3,1), ss(3,2), ss(3,3); ...
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LINKS
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FORMULA
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c satisfies the recurrence c(m, k)=1/m sum_{j=1..m} (((k+1) j-m)c(m-j, k)c(j, 1)), where c(0, k)=1 and c(j, 1)= (1/2) (-1)^j 1/(j+1)! sum_{i=1..j+1} binomial(j+1, i) i^j.
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EXAMPLE
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2; 4,8; 24,48,48; 224,480,576,384; ...
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MATHEMATICA
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(* ccctri lists first numrows rows of triangular array. *)
ccctri[numrows_] := (s[j_] := Sum[Binomial[j, i] i^(j-1), {i, 1, j}]; r[j_] := 1/2 (-1)^j 1/(j+1)! s[j+1]; c[m_, k_] := 1/m Sum[((k+1) j-m)c[m-j, k]r[j], {j, 1, m}]; c[0, k_] := 1; ss[m_, k_] := 2^k m! (-1)^(m-k) c[m-k, k]; Flatten[Table[Table[ss[k, j], {j, 1, k}], {k, 1, numrows}]])
(* ccccol lists maxrow elements of column colnum. *)
ccccol[colnum_, maxrow_] := (s[j_] := Sum[Binomial[j, i] i^(j-1), {i, 1, j}]; r[j_] := 1/2 (-1)^j 1/(j+1)! s[j+1]; c[m_, k_] := 1/m Sum[((k+1) j-m)c[m-j, k]r[j], {j, 1, m}]; c[0, k_] := 1; ss[m_, k_] := 2^k m! (-1)^(m-k) c[m-k, k]; Table[ss[m, colnum], {m, colnum, maxrow}])
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CROSSREFS
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First column of triangular array (ss(m, 1) for m>=1) is A038049.
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KEYWORD
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AUTHOR
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Carmen Chicone (carmen(AT)chicone.math.missouri.edu), Dec 22 2002
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EXTENSIONS
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STATUS
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approved
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