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 A237765 Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n. 0
 0, 0, 0, 1, 2, 1, 3, 9, 9, 3, 6, 24, 36, 24, 6, 10, 50, 100, 100, 50, 10, 15, 90, 225, 300, 225, 90, 15, 21, 147, 441, 735, 735, 441, 147, 21, 28, 224, 784, 1568, 1960, 1568, 784, 224, 28, 36, 324, 1296, 3024, 4536, 4536, 3024, 1296, 324, 36 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS T(n,k) is the number of ways to underline exactly two elements of {1,2,...,n} and then circle exactly k elements. (The k elements that are circled are not necessarily different from the two underlined elements). T(n,0) = T(n,n) = binomial(n,2) = A000217(n-1). Row sums = 2^n*binomial(n,2) = A100381(n). REFERENCES J. Riordan, Introduction to Combinatorial Analysis, Wiley, 1958, page 14, problem #2. LINKS FORMULA E.g.f.: (x^2/2! + 2*y*x^2/2! + y^2*x^2/2!)*exp(y*x)*exp(x). E.g.f. for column k: x^2/2!*exp(x)*(x^k/k! + 2*x^(k-1)/(k-1)! + x^(k-2)/(k-2)!). T(n,k) = C(n,2)*( C(n-2,k) + 2*C(n-2,k-1) + C(n-2,k-2) ). EXAMPLE 0; 0,  0; 1,  2,   1; 3,  9,   9,    3; 6,  24,  36,   24,   6; 10, 50,  100,  100,  50,   10; 15, 90,  225,  300,  225,  90,   15; 21, 147, 441,  735,  735,  441,  147,  21; 28, 224, 784,  1568, 1960, 1568, 784,  224,  28; 36, 324, 1296, 3024, 4536, 4536, 3024, 1296, 324, 36; MATHEMATICA Table[Table[Binomial[n, 2](Binomial[n-2, r]+2Binomial[n-2, r-1]+Binomial[n-2, r-2]), {r, 0, n}], {n, 0, 9}]//Grid CROSSREFS Cf. A134400. Sequence in context: A058113 A249456 A234746 * A249632 A126009 A301282 Adjacent sequences:  A237762 A237763 A237764 * A237766 A237767 A237768 KEYWORD nonn,tabl AUTHOR Geoffrey Critzer, Feb 12 2014 STATUS approved

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Last modified August 12 00:34 EDT 2022. Contains 356067 sequences. (Running on oeis4.)