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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Adjacent sequences:  A253270 A253271 A253272 * A253274 A253275 A253276

KEYWORD

nonn,tabl

AUTHOR

Vladimir Kruchinin, May 01 2015

STATUS

approved

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Last modified June 14 21:27 EDT 2021. Contains 345041 sequences. (Running on oeis4.)