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A156052 Triangle read by rows: T(n, k) = binomial(n, k)/Beta(n+1, n-k+1) + binomial(n, n-k)/Beta(n+1, k+1). 1
2, 8, 8, 33, 48, 33, 144, 240, 240, 144, 635, 1240, 1260, 1240, 635, 2778, 6510, 6720, 6720, 6510, 2778, 12019, 33600, 38430, 33600, 38430, 33600, 12019, 51488, 168672, 223776, 184800, 184800, 223776, 168672, 51488, 218799, 824400, 1275120, 1119888, 900900, 1119888, 1275120, 824400, 218799 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Row sums are 2*A108666(n+1): {2, 16, 114, 768, 5010, 32016, 201698, 1257472, 7777314, 47800080, 292292946, ...}.

LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened

FORMULA

T(n, k) = binomial(n, k)/Beta(n+1, n-k+1) + binomial(n, n-k)/Beta(n+1, k+1).

T(n, k) = (k+1)*binomial(n+1, k+1)*( binomial(2*n-k+1, n+1) + binomial(n+k+1, n+1) ). - G. C. Greubel, Dec 01 2019

EXAMPLE

Triangle begins as:

2;

8, 8;

33, 48, 33;

144, 240, 240, 144;

635, 1240, 1260, 1240, 635;

2778, 6510, 6720, 6720, 6510, 2778;

12019, 33600, 38430, 33600, 38430, 33600, 12019;

51488, 168672, 223776, 184800, 184800, 223776, 168672, 51488;

MAPLE

b:=binomial; seq(seq( (k+1)*b(n+1, k+1)*( b(2*n-k+1, n+1) + b(n+k+1, n+1) ), k=0..n), n=0..10); # G. C. Greubel, Dec 01 2019

MATHEMATICA

Table[Binomial[n, k]/Beta[n+1, n-k+1] + Binomial[n, n-k]/Beta[n+1, k+1], {n, 0, 10}, {k, 0, n}]//FlattenTable[(k+1)*Binomial[n+1, k+1]*(Binomial[n+k+1, n+1] + Binomial[2*n-k+1, n+1]), {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 01 2019 *)

PROG

(PARI) T(n, k) = my(b=binomial); (k+1)*b(n+1, k+1)*( b(2*n-k+1, n+1) + b(n+k+1, n+1) ); \\ G. C. Greubel, Dec 01 2019

(Magma) B:=Binomial; [(k+1)*B(n+1, k+1)*( B(2*n-k+1, n+1) + B(n+k+1, n+1) ): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 01 2019

(Sage) b=binomial; [[(k+1)*b(n+1, k+1)*( b(2*n-k+1, n+1) + b(n+k+1, n+1) ) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 01 2019

(GAP) B:=Binomial;; Flat(List([0..10], n-> List([0..n], k-> (k+1)*B(n+1, k+1)*( B(2*n-k+1, n+1) + B(n+k+1, n+1) ) ))); # G. C. Greubel, Dec 01 2019

CROSSREFS

Sequence in context: A064231 A181130 A212196 * A170923 A083523 A202619

Adjacent sequences: A156049 A156050 A156051 * A156053 A156054 A156055

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 02 2009

STATUS

approved

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Last modified February 9 07:25 EST 2023. Contains 360153 sequences. (Running on oeis4.)