A210554 Triangle of coefficients of polynomials v(n,x) jointly generated with A208341; see the Formula section.
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, 144, 64, 8, 35, 104, 225, 360, 416, 320, 128, 9, 44, 147, 363, 681, 968, 1008, 704, 256, 10, 54, 200, 553, 1182, 1970, 2528, 2400, 1536, 512
Offset: 1
Examples
Triangle begins: 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, 144, 64; ... First three polynomials v(n,x): 1, 2 + 2x , 3 + 5x + 4x^2. The T(3, 1) = 3 multisets: (1), (2), (3). The T(3, 2) = 5 multisets: (11), (12), (13), (22), (23). The T(3, 3) = 4 multisets: (111), (112), (122), (123).
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275
Programs
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Maple
T := (n,k) -> simplify((n + 1 - k)*hypergeom([1 - k, -k + n + 2], [2], -1)): seq(seq(T(n,k), k=1..n), n=1..10); # Peter Luschny, Sep 18 2018
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Mathematica
u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1; v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1; Table[Expand[u[n, x]], {n, 1, z/2}] Table[Expand[v[n, x]], {n, 1, z/2}] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu] Flatten[%] (* A208341 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A210554 *) -
PARI
T(n,k)={sum(i=1, k, binomial(k-1, i-1)*binomial(n-k+i, i))} \\ Andrew Howroyd, Sep 18 2018
Formula
u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
T(n,k) = Sum_{i=1..k} binomial(k-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018
T(n,k) = (n - k + 1)*hypergeom([1 - k, n - k + 2], [2], -1). - Peter Luschny, Sep 18 2018
Extensions
Example corrected by Philippe Deléham, Mar 23 2012
Comments