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A024996
Triangular array, read by rows: second differences in n,n direction of trinomial array A027907.
18
1, 1, 0, 1, 1, 0, 2, 0, 1, 1, 1, 3, 2, 3, 1, 1, 1, 2, 5, 6, 8, 6, 5, 2, 1, 1, 3, 8, 13, 19, 20, 19, 13, 8, 3, 1, 1, 4, 12, 24, 40, 52, 58, 52, 40, 24, 12, 4, 1, 1, 5, 17, 40, 76, 116, 150, 162, 150, 116, 76, 40, 17, 5, 1, 1, 6, 23, 62, 133, 232, 342, 428, 462, 428, 342, 232, 133, 62, 23, 6
OFFSET
0,7
COMMENTS
For n > 2, T(n,k) is the number of integer strings s(0), ..., s(n) such that s(n) = n - k, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2 and <= 1 for i >= 3.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1874 [a(676) ff. corrected by Georg Fischer, Jun 24 2020]
FORMULA
T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 0, 1], [1, 0, 2, 0, 1].
G.f.: y*z + (1-y*z)^2 / (1-z*(1+y+y^2)). - Ralf Stephan, Jan 09 2005 [corrected by Peter Luschny, Jun 25 2020]
EXAMPLE
1
1 0 1
1 0 2 0 1
1 1 3 2 3 1 1
1 2 5 6 8 6 5 2 1
1 3 8 13 19 20 19 13 8 3 1
MAPLE
A024996 := proc(n, k)
option remember;
if n < 0 or k < 0 or k > 2*n then
0 ;
elif n <= 2 then
if k = 2*n or k = 0 then
1;
elif k = 2*n-1 or k = 1 then
0;
elif k =2 then
2;
end if;
else
procname(n-1, k-1)+procname(n-1, k-2)+procname(n-1, k) ;
end if;
end proc: # R. J. Mathar, Jun 23 2013
seq(seq(A024996(n, k), k=0..2*n), n=0..11); # added by Georg Fischer, Jun 24 2020
MATHEMATICA
nmax = 10; CoefficientList[CoefficientList[Series[y*x + (1 - y*x)^2/(1 - x*(1 + y + y^2)), {x, 0, nmax}, {y, 0, 2*nmax}], x], y] // Flatten (* G. C. Greubel, May 22 2017; amended by Georg Fischer, Jun 24 2020 *)
PROG
(PARI) T(n, k)=if(n<0||k<0||k>2*n, 0, if(n==0, 1, if(n==1, [1, 0, 1][k+1], if(n==2, [1, 0, 2, 0, 1][k+1], T(n-1, k-2)+T(n-1, k-1)+T(n-1, k))))) \\ Ralf Stephan, Jan 09 2004
nmax=8; for(n=0, nmax, for(k=0, 2*n, print1(T(n, k), ", "))) \\ added by _Georg Fischer, Jun 24 2020
(Julia)
using Nemo
function A024996Expansion(prec)
R, t = PolynomialRing(ZZ, "t")
S, x = PowerSeriesRing(R, prec+1, "x")
ser = divexact(x^2*t^3 + x^2*t + x*t - 1, x*t^2 + x*t + x - 1)
L = zeros(ZZ, prec^2)
for k ∈ 0:prec-1, n ∈ 0:2*k
L[k^2+n+1] = coeff(coeff(ser, k), n)
end
L
end
A024996Expansion(8) |> println # Peter Luschny, Jun 25 2020
CROSSREFS
First differences in n, n direction of array A025177.
Central column is essentially A024997, other columns are A024998, A026069, A026070, A026071. Row sums are in A025579.
Sequence in context: A357914 A035467 A254045 * A187596 A263863 A134655
KEYWORD
nonn,tabf,easy
EXTENSIONS
Edited by Ralf Stephan, Jan 09 2004
Offset corrected by R. J. Mathar, Jun 23 2013
STATUS
approved