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A211563 Number of nonnegative integer arrays of length n+3 with new values 0 upwards introduced in order, and containing the value n-1.

%I A211563 #8 Jul 19 2018 05:50:02
%S A211563 15,51,171,512,1345,3145,6676,13091,24047,41835,69525,111126,171761,
%T A211563 257857,377350,539905,757151,1042931,1413567,1888140,2488785,3241001,
%U A211563 4173976,5320927,6719455,8411915,10445801,12874146,15755937,19156545
%N A211563 Number of nonnegative integer arrays of length n+3 with new values 0 upwards introduced in order, and containing the value n-1.
%C A211563 Row 4 of A211561.
%H A211563 R. H. Hardin, <a href="/A211563/b211563.txt">Table of n, a(n) for n = 1..210</a>
%F A211563 Empirical: a(n) = (1/48)*n^6 + (7/48)*n^5 + (23/48)*n^4 + (61/48)*n^3 + 3*n^2 + (61/12)*n + 5.
%F A211563 Empirical: a(n) = sum{j in n..n+3}stirling2(n+3,j).
%F A211563 Conjectures from _Colin Barker_, Jul 19 2018: (Start)
%F A211563 G.f.: x*(15 - 54*x + 129*x^2 - 139*x^3 + 92*x^4 - 33*x^5 + 5*x^6) / (1 - x)^7.
%F A211563 a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
%F A211563 (End)
%e A211563 Some solutions for n=5:
%e A211563 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A211563 ..1....1....1....1....0....1....1....1....1....1....1....1....1....1....1....1
%e A211563 ..2....2....2....2....1....0....2....2....2....2....2....2....2....2....2....2
%e A211563 ..1....3....3....3....2....2....3....3....2....0....3....0....3....1....2....1
%e A211563 ..3....4....2....0....3....3....4....0....1....3....4....0....0....1....3....3
%e A211563 ..3....5....4....3....4....4....5....4....3....4....4....3....2....0....4....4
%e A211563 ..4....6....4....3....1....2....6....2....2....4....5....4....4....3....0....4
%e A211563 ..1....1....1....4....5....1....7....0....4....0....5....4....1....4....3....1
%Y A211563 Cf. A211561.
%K A211563 nonn
%O A211563 1,1
%A A211563 _R. H. Hardin_, Apr 15 2012