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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
OFFSET
0,1
REFERENCES
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 66.
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
Sequence in context: A167886 A095978 A370361 * A169653 A129012 A136122
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 15 2009
EXTENSIONS
Edited by G. C. Greubel, Jun 15 2021
STATUS
approved