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A171822 Triangle T(n,k) = binomial(2*n-k, k)*binomial(n+k, 2*k), read by rows. 2
1, 1, 1, 1, 9, 1, 1, 30, 30, 1, 1, 70, 225, 70, 1, 1, 135, 980, 980, 135, 1, 1, 231, 3150, 7056, 3150, 231, 1, 1, 364, 8316, 34650, 34650, 8316, 364, 1, 1, 540, 19110, 132132, 245025, 132132, 19110, 540, 1, 1, 765, 39600, 420420, 1288287, 1288287, 420420, 39600, 765, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

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

FORMULA

T(n, k) = binomial(2*n-k, k)*binomial(n+k, 2*k) = A054142(n, k)*A085478(n, k).

Sum_{k=0..n} T(n, k) = Hypergeometric 4F3([-n, -n, 1/2 -n, n+1], [1/2, 1, -2*n], 1) = A183160(n). - G. C. Greubel, Feb 22 2021

EXAMPLE

Triangle begins as:

  1;

  1,    1;

  1,    9,     1;

  1,   30,    30,       1;

  1,   70,   225,      70,       1;

  1,  135,   980,     980,     135,       1;

  1,  231,  3150,    7056,    3150,     231,       1;

  1,  364,  8316,   34650,   34650,    8316,     364,       1;

  1,  540, 19110,  132132,  245025,  132132,   19110,     540,     1;

  1,  765, 39600,  420420, 1288287, 1288287,  420420,   39600,   765,    1;

  1, 1045, 75735, 1166880, 5465460, 9018009, 5465460, 1166880, 75735, 1045, 1;

MATHEMATICA

Table[Binomial[2*n-k, k]*Binomial[n+k, 2*k], {n, 0, 10}, {k, 0, n}]//Flatten

PROG

(Sage) flatten([[binomial(2*n-k, k)*binomial(n+k, 2*k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 22 2021

(Magma) [Binomial(2*n-k, k)*Binomial(n+k, 2*k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 22 2021

CROSSREFS

Cf. A054142, A085478, A183160.

Sequence in context: A144404 A014761 A073702 * A176490 A174158 A181144

Adjacent sequences:  A171819 A171820 A171821 * A171823 A171824 A171825

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Dec 19 2009

EXTENSIONS

Edited by G. C. Greubel, Feb 22 2021

STATUS

approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)