A156763 - OEIS

%I #11 Sep 08 2022 08:45:41

%S 2,3,3,7,12,7,21,42,42,21,71,160,180,160,71,253,660,770,770,660,253,

%T 925,2814,3570,3360,3570,2814,925,3433,12068,17388,15750,15750,17388,

%U 12068,3433,12871,51552,85344,81312,69300,81312,85344,51552,12871

%N 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.

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

%H G. C. Greubel, <a href="/A156763/b156763.txt">Rows n = 0..50 of the triangle, flattened</a>

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

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

%F Sum_{k=0..n} T(n, k) = 2*A001850(n). - _G. C. Greubel_, Jun 15 2021

%e Triangle begins as:

%e       2;

%e       3,      3;

%e       7,     12,      7;

%e      21,     42,     42,     21;

%e      71,    160,    180,    160,     71;

%e     253,    660,    770,    770,    660,    253;

%e     925,   2814,   3570,   3360,   3570,   2814,    925;

%e    3433,  12068,  17388,  15750,  15750,  17388,  12068,   3433;

%e   12871,  51552,  85344,  81312,  69300,  81312,  85344,  51552,  12871;

%e   48621, 218880, 413820, 438900, 342342, 342342, 438900, 413820, 218880, 48621;

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

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 15 2021 *)

%o (Magma)

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

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

%o [A156763(n,k): k in [0..n]. n in [0..12]]; // _G. C. Greubel_, Jun 15 2021

%o (Sage)

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

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

%o flatten([[A156763(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 15 2021

%Y Cf. A001850, A063007.

%K nonn,tabl

%O 0,1

%A _Roger L. Bagula_, Feb 15 2009

%E Edited by _G. C. Greubel_, Jun 15 2021