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A024996
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Triangular array, read by rows: second differences in n,n direction of trinomial array A027907.
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18
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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
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internal format)
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OFFSET
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0,7
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COMMENTS
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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.
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LINKS
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FORMULA
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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].
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EXAMPLE
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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
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MAPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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First differences in n, n direction of array A025177.
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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