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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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21
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1, 4, 20, 116, 756, 5428, 42356, 355636, 3188340, 30333492, 304716148, 3218555700, 35618229364, 411717043252, 4957730174836, 62045057731892, 805323357485684, 10820999695801908, 150271018666120564, 2153476417340487476
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Also, 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 (pauldhanna(AT)juno.com), Dec 08 2003
First column of PE^4, where PE is given in A011971, second power in A078937, third power in A078938, fourth power in A078939 - Gottfried Helms (helms(AT)uni-kassel.de), Apr 08 2007
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FORMULA
| 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 (helms(AT)uni-kassel.de), Apr 08 2007
E.g.f.: exp{4(e^x-1)}.
a(n) = exp(-4)*sum(k>=0, 4^k*k^n/k! ) - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 25 2003
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 (pauldhanna(AT)juno.com), Dec 08 2003
a(n) = Sum_{k = 0..n} 4^k*A048993(n, k); A048993 : Stirling-2 numbers . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), May 09 2004
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MAPLE
| 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 (mathar(AT)strw.leidenuniv.nl), May 30 2008
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MATHEMATICA
| Table[n!, {n, 0, 20}]CoefficientList[Series[E^(4E^x-4), {x, 0, 20}], x]
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PROG
| sage: from sage.combinat.expnums import expnums2 sage: expnums(20, 4) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008
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CROSSREFS
| Cf. A078939, A001861, A056857, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.
Sequence in context: A100328 A082298 A129378 * A158900 A190194 A127088
Adjacent sequences: A078941 A078942 A078943 * A078945 A078946 A078947
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Dec 18 2002
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 30 2008
Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 02 2008 at the suggestion of R. J. Mathar
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