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%I #18 Sep 18 2018 17:00:38
%S 1,2,2,3,5,4,4,9,12,8,5,14,25,28,16,6,20,44,66,64,32,7,27,70,129,168,
%T 144,64,8,35,104,225,360,416,320,128,9,44,147,363,681,968,1008,704,
%U 256,10,54,200,553,1182,1970,2528,2400,1536,512
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A208341; see the Formula section.
%C For a discussion and guide to related arrays, see A208510.
%C Also the number of multisets of size k that fit within some normal multiset of size n. A multiset is normal if it spans an initial interval of positive integers. - _Andrew Howroyd_, Sep 18 2018
%H Andrew Howroyd, <a href="/A210554/b210554.txt">Table of n, a(n) for n = 1..1275</a>
%F u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
%F v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F where u(1,x)=1, v(1,x)=1.
%F T(n,k) = Sum_{i=1..k} binomial(k-1, i-1)*binomial(n-k+i, i). - _Andrew Howroyd_, Sep 18 2018
%F T(n,k) = (n - k + 1)*hypergeom([1 - k, n - k + 2], [2], -1). - _Peter Luschny_, Sep 18 2018
%e Triangle begins:
%e 1;
%e 2, 2;
%e 3, 5, 4;
%e 4, 9, 12, 8;
%e 5, 14, 25, 28, 16;
%e 6, 20, 44, 66, 64, 32;
%e 7, 27, 70, 129, 168, 144, 64;
%e ...
%e First three polynomials v(n,x): 1, 2 + 2x , 3 + 5x + 4x^2.
%e The T(3, 1) = 3 multisets: (1), (2), (3).
%e The T(3, 2) = 5 multisets: (11), (12), (13), (22), (23).
%e The T(3, 3) = 4 multisets: (111), (112), (122), (123).
%p T := (n,k) -> simplify((n + 1 - k)*hypergeom([1 - k, -k + n + 2], [2], -1)):
%p seq(seq(T(n,k), k=1..n), n=1..10); # _Peter Luschny_, Sep 18 2018
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A208341 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A210554 *)
%o (PARI) T(n,k)={sum(i=1, k, binomial(k-1, i-1)*binomial(n-k+i, i))} \\ _Andrew Howroyd_, Sep 18 2018
%Y Row sums are A027941.
%Y Cf. A160232, A208341, A208510, A303974.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Mar 22 2012
%E Example corrected by _Philippe Deléham_, Mar 23 2012
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