This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A210554 #18 Sep 18 2018 17:00:38
%S A210554 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 A210554 144,64,8,35,104,225,360,416,320,128,9,44,147,363,681,968,1008,704,
%U A210554 256,10,54,200,553,1182,1970,2528,2400,1536,512
%N A210554 Triangle of coefficients of polynomials v(n,x) jointly generated with A208341; see the Formula section.
%C A210554 For a discussion and guide to related arrays, see A208510.
%C A210554 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 A210554 Andrew Howroyd, <a href="/A210554/b210554.txt">Table of n, a(n) for n = 1..1275</a>
%F A210554 u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
%F A210554 v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A210554 where u(1,x)=1, v(1,x)=1.
%F A210554 T(n,k) = Sum_{i=1..k} binomial(k-1, i-1)*binomial(n-k+i, i). - _Andrew Howroyd_, Sep 18 2018
%F A210554 T(n,k) = (n - k + 1)*hypergeom([1 - k, n - k + 2], [2], -1). - _Peter Luschny_, Sep 18 2018
%e A210554 Triangle begins:
%e A210554 1;
%e A210554 2, 2;
%e A210554 3, 5, 4;
%e A210554 4, 9, 12, 8;
%e A210554 5, 14, 25, 28, 16;
%e A210554 6, 20, 44, 66, 64, 32;
%e A210554 7, 27, 70, 129, 168, 144, 64;
%e A210554 ...
%e A210554 First three polynomials v(n,x): 1, 2 + 2x , 3 + 5x + 4x^2.
%e A210554 The T(3, 1) = 3 multisets: (1), (2), (3).
%e A210554 The T(3, 2) = 5 multisets: (11), (12), (13), (22), (23).
%e A210554 The T(3, 3) = 4 multisets: (111), (112), (122), (123).
%p A210554 T := (n,k) -> simplify((n + 1 - k)*hypergeom([1 - k, -k + n + 2], [2], -1)):
%p A210554 seq(seq(T(n,k), k=1..n), n=1..10); # _Peter Luschny_, Sep 18 2018
%t A210554 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210554 u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210554 v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A210554 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210554 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210554 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210554 TableForm[cu]
%t A210554 Flatten[%] (* A208341 *)
%t A210554 Table[Expand[v[n, x]], {n, 1, z}]
%t A210554 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210554 TableForm[cv]
%t A210554 Flatten[%] (* A210554 *)
%o A210554 (PARI) T(n,k)={sum(i=1, k, binomial(k-1, i-1)*binomial(n-k+i, i))} \\ _Andrew Howroyd_, Sep 18 2018
%Y A210554 Row sums are A027941.
%Y A210554 Cf. A160232, A208341, A208510, A303974.
%K A210554 nonn,tabl
%O A210554 1,2
%A A210554 _Clark Kimberling_, Mar 22 2012
%E A210554 Example corrected by _Philippe Deléham_, Mar 23 2012