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A078944
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First column of A078939, the fourth power of lower triangular matrix A056857.
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35
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1, 4, 20, 116, 756, 5428, 42356, 355636, 3188340, 30333492, 304716148, 3218555700, 35618229364, 411717043252, 4957730174836, 62045057731892, 805323357485684, 10820999695801908, 150271018666120564, 2153476417340487476
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OFFSET
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0,2
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COMMENTS
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Also, the number of ways of placing n labeled balls into n unlabeled (but 4-colored) boxes. Binomial transform of this sequence is A078945 and a(n+1) = 4*A078945(n). - Paul D. Hanna, Dec 08 2003
The number of ways of putting n labeled balls into a set of bags and then putting the bags into 4 labeled boxes. - Peter Bala, Mar 23 2013
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LINKS
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FORMULA
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PE=exp(matpascal(5))/exp(1); A = PE^4; a(n)= A[ n,1 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^4; a(n)=A[ n,1]. - Gottfried Helms, Apr 08 2007
E.g.f.: exp(4*(exp(x)-1)).
G.f.: 4*(x/(1-x))*A(x/(1-x)) = A(x) - 1; four times the binomial transform equals this sequence shifted one place left. - Paul D. Hanna, Dec 08 2003
G.f.: (G(0) - 1)/(x-1)/4 where G(k) = 1 - 4/(1-k*x)/(1-x/(x-1/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: T(0)/(1-4*x), where T(k) = 1 - 4*x^2*(k+1)/(4*x^2*(k+1) - (1-(k+4)*x)*(1-(k+5)*x)/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 28 2013
a(n) ~ n^n * exp(n/LambertW(n/4)-4-n) / (sqrt(1+LambertW(n/4)) * LambertW(n/4)^n). - Vaclav Kotesovec, Mar 12 2014
G.f.: Sum_{j>=0} 4^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 07 2019
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MAPLE
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A056857 := proc(n, c) combinat[bell](n-1-c)*binomial(n-1, c) ; end: A078937 := proc(n, c) add( A056857(n, k)*A056857(k+1, c), k=0..n) ; end: A078938 := proc(n, c) add( A078937(n, k)*A056857(k+1, c), k=0..n) ; end: A078939 := proc(n, c) add( A078938(n, k)*A056857(k+1, c), k=0..n) ; end: A078944 := proc(n) A078939(n+1, 0) ; end: seq(A078944(n), n=0..25) ; # R. J. Mathar, May 30 2008
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0, 4^m,
add(b(n-1, max(m, j)), j=1..m+1))
end:
a:= n-> b(n, 0):
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MATHEMATICA
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Table[n!, {n, 0, 20}]CoefficientList[Series[E^(4E^x-4), {x, 0, 20}], x]
With[{nn=20}, CoefficientList[Series[Exp[4(Exp[x]-1)], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, May 03 2022 *)
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PROG
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CROSSREFS
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Cf. A000110, A001861, A027710, A056857, A078937, A078938, A078939, A078944, A078945, A129323, A129324, A129325, A129327, A129328, A129329, A129331, A129332, A129333, A144180, A144223, A144263, A189233, A221159, A221176.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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