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A229675
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a(n) = Sum_{k = 0..n} Product_{j = 0..7} C(n+j*k,k).
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3
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1, 40321, 81730010881, 369400348294790401, 2390471064720364776796801, 18975660656355118819906214670721, 171890067585060168829713844899790066561, 1707759022485971054271963683059722310362986881, 18165821273625565354157327818616137066973745155992321
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OFFSET
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0,2
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COMMENTS
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Number of lattice paths from {n}^8 to {0}^8 using steps that decrement one component or all components by 1.
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LINKS
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FORMULA
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a(n) = Sum_{k = 0..n} multinomial(n+7*k; n-k, {k}^8).
G.f.: Sum_{k >= 0} (8*k)!/k!^8 * x^k / (1-x)^(8*k+1).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 20161*x^2 + 27243357121*x^3 + 92350114520267521*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016
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MAPLE
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with(combinat):
a:= n-> add(multinomial(n+7*k, n-k, k$8), k=0..n):
seq(a(n), n=0..10);
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MATHEMATICA
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multinomial[n_, k_List] := n!/Times @@ (k!); a[n_] := Sum[multinomial[n + 7*k, Join[{n - k}, Array[k&, 8]]], {k, 0, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Dec 27 2013, translated from Maple *)
Table[Sum[Product[Binomial[n+j*k, k], {j, 0, 7}], {k, 0, n}], {n, 0, 10}] (* Harvey P. Dale, Aug 25 2014 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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