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A156763 Triangle T(n, k) = binomial(2*k, k)*binomial(n+k, n-k) + binomial(2*n-k, k)*binomial(2*(n-k), n-k), read by rows. 1
2, 3, 3, 7, 12, 7, 21, 42, 42, 21, 71, 160, 180, 160, 71, 253, 660, 770, 770, 660, 253, 925, 2814, 3570, 3360, 3570, 2814, 925, 3433, 12068, 17388, 15750, 15750, 17388, 12068, 3433, 12871, 51552, 85344, 81312, 69300, 81312, 85344, 51552, 12871 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

REFERENCES

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 66.

LINKS

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

FORMULA

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

T(n, k) = A063007(n, k) + A063007(n, n-k).

Sum_{k=0..n} T(n, k) = 2*A001850(n). - G. C. Greubel, Jun 15 2021

EXAMPLE

Triangle begins as:

2;

3, 3;

7, 12, 7;

21, 42, 42, 21;

71, 160, 180, 160, 71;

253, 660, 770, 770, 660, 253;

925, 2814, 3570, 3360, 3570, 2814, 925;

3433, 12068, 17388, 15750, 15750, 17388, 12068, 3433;

12871, 51552, 85344, 81312, 69300, 81312, 85344, 51552, 12871;

48621, 218880, 413820, 438900, 342342, 342342, 438900, 413820, 218880, 48621;

MATHEMATICA

T[n_, k_]:= Binomial[n+k, n-k]*Binomial[2*k, k] + Binomial[2*(n-k), n-k]*Binomial[ 2*n-k, k];

Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 15 2021 *)

PROG

(Magma)

A063007:= func< n, k | Binomial(n, k)*Binomial(n+k, k) >;

A156763:= func< n, k | A063007(n, k) + A063007(n, n-k) >;

[A156763(n, k): k in [0..n]. n in [0..12]]; // G. C. Greubel, Jun 15 2021

(Sage)

def A063007(n, k): return binomial(n+k, n-k)*binomial(2*k, k)

def A156763(n, k): return A063007(n, k) + A063007(n, n-k)

flatten([[A156763(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 15 2021

CROSSREFS

Cf. A001850, A063007.

Sequence in context: A098715 A167886 A095978 * A169653 A129012 A136122

Adjacent sequences: A156760 A156761 A156762 * A156764 A156765 A156766

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 15 2009

EXTENSIONS

Edited by G. C. Greubel, Jun 15 2021

STATUS

approved

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Last modified December 6 09:59 EST 2022. Contains 358622 sequences. (Running on oeis4.)