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A119673
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T(n, k) = 3*T(n-1, k-1) + T(n-1, k) for k < n and T(n, n) = 1, T(n, k) = 0, if k < 0 or k > n; triangle read by rows.
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2
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1, 1, 1, 1, 4, 1, 1, 7, 13, 1, 1, 10, 34, 40, 1, 1, 13, 64, 142, 121, 1, 1, 16, 103, 334, 547, 364, 1, 1, 19, 151, 643, 1549, 2005, 1093, 1, 1, 22, 208, 1096, 3478, 6652, 7108, 3280, 1, 1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1, 1, 28, 349, 2542, 11926, 37384, 78322, 105796, 83653, 29524, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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LINKS
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G. C. Greubel, Rows n = 0..100 of triangle, flattened
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FORMULA
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T(n,k) = R(n,k,3) where R(n,k,m) = (1-m)^(-n+k)-m^(k+1)*Pochhammer(n-k, k+1)* hyper2F1([1,n+1],[k+2],m)/(k+1)!. - Peter Luschny, Jul 25 2014
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 4, 1;
1, 7, 13, 1;
1, 10, 34, 40, 1;
1, 13, 64, 142, 121, 1;
1, 16, 103, 334, 547, 364, 1;
1, 19, 151, 643, 1549, 2005, 1093, 1;
1, 22, 208, 1096, 3478, 6652, 7108, 3280, 1;
1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1;
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MAPLE
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T := (n, k, m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k, k+1)* hypergeom([1, n+1], [k+2], m)/(k+1)!; A119673 := (n, k) -> T(n, k, 3);
seq(print(seq(round(evalf(A119673(n, k))), k=0..n)), n=0..10); # Peter Luschny, Jul 25 2014
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MATHEMATICA
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T[_, 0]=1; T[n_, n_]=1; T[n_, k_]/; 0<k<n := T[n, k] = 3T[n-1, k-1] + T[n-1, k]; T[_, _] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)
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PROG
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(PARI) T(n, k) = if(k<0 || k>n, 0, if(k==n, 1, 3*T(n-1, k-1) +T(n-1, k)));
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Nov 18 2019
(Magma)
function T(n, k)
if k lt 0 or k gt n then return 0;
elif k eq n then return 1;
else return 3*T(n-1, k-1) + T(n-1, k);
end if;
return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 18 2019
(Sage)
@CachedFunction
def T(n, k):
if (k<0 or k>n): return 0
elif (k==n): return 1
else: return 3*T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 18 2019
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CROSSREFS
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Cf. A003462, A014915, A081271, A119258.
Sequence in context: A209414 A193636 A232968 * A144447 A051455 A346875
Adjacent sequences: A119670 A119671 A119672 * A119674 A119675 A119676
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Zerinvary Lajos, Jun 11 2006
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EXTENSIONS
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Definition clarified by Philippe Deléham, Jun 13 2006
Entry revised by N. J. A. Sloane, Jun 19 2006
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STATUS
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approved
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