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A253273
Triangle T(n,k) = Sum_{j=0..n-k+1} binomial(k+j,k-j+1)*binomial(n-k,j-1), read by rows.
1
1, 1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 12, 14, 5, 1, 6, 18, 30, 25, 6, 1, 7, 25, 53, 66, 41, 7, 1, 8, 33, 84, 136, 132, 63, 8, 1, 9, 42, 124, 244, 315, 245, 92, 9, 1, 10, 52, 174, 400, 636, 673, 428, 129, 10, 1, 11, 63, 235, 615, 1152, 1522, 1346, 711, 175, 11
OFFSET
0,3
FORMULA
T(n,k) = Sum_{j=0..n-k+1} binomial(k+j,k-j+1)*binomial(n-k,j-1).
Sum_{k=0..n} T(n,k) = A095263(n+1).
G.f.: 1/( (1-x)*(1+y^2) - (2-x)*y ).
EXAMPLE
The triangle begins as:
1;
1, 2;
1, 3, 3;
1, 4, 7, 4;
1, 5, 12, 14, 5;
1, 6, 18, 30, 25, 6;
1, 7, 25, 53, 66, 41, 7;
1, 8, 33, 84, 136, 132, 63, 8;
1, 9, 42, 124, 244, 315, 245, 92, 9;
1, 10, 52, 174, 400, 636, 673, 428, 129, 10;
...
MATHEMATICA
T[n_, k_]:= Sum[Binomial[k+j, k-j+1]*Binomial[n-k, j-1], {j, 0, n-k+1}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 17 2021 *)
PROG
(Maxima)
T(n, m):=sum(binomial(m+k, m-k+1)*binomial(n-m, k-1), k, 0, n-m+1);
(Magma)
T:= func< n, k | (&+[Binomial(k+j, k-j+1)*Binomial(n-k, j-1): j in [0..n-k+1]]) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 17 2021
(Sage)
def T(n, k): return sum(binomial(k+j, k-j+1)*binomial(n-k, j-1) for j in (0..n-k+1))
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 17 2021
CROSSREFS
Cf. A095263.
Sequence in context: A319539 A098546 A126277 * A055129 A133804 A185943
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, May 01 2015
STATUS
approved