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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.)