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 A039597 Triangle read by rows: T(n,k) = number of 2 X inf arrays [ n, n1, n2, ...; k, k1, k2,... ] with n>=n1>n2>...>=0, k>=k1>k2...>=0, n>k, n1>k1, ...; n >= 1, k >= 0. Note that once ni or ki = 0, the strict inequalities become equalities (constant 0 thereafter). 3
 2, 4, 6, 8, 14, 20, 16, 30, 50, 70, 32, 62, 112, 182, 252, 64, 126, 238, 420, 672, 924, 128, 254, 492, 912, 1584, 2508, 3432, 256, 510, 1002, 1914, 3498, 6006, 9438, 12870, 512, 1022, 2024, 3938, 7436, 13442, 22880, 35750, 48620, 1024, 2046, 4070, 8008, 15444 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 REFERENCES L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 163. LINKS FORMULA Left column = powers of 2, right column = central binomial coefficients C(2n,n); interior entries = sum of entry above and entry to the left. - N. J. A. Sloane, Jun 18 2016 EXAMPLE T(2,1) = 6 because the n row can go 2,2,1,0 with the k row either 1,1,0,0 or 1,0,0,0; the n row can go 2,2,0,0 with those same k rows; or the n row can go 2,1,0,0 or 2,0,0,0 with the k row going 1,0,0,0 (since k must be strictly less than n, except when both are 0). Triangle begins: 2, 4, 6, 8, 14, 20, 16, 30, 50, 70, 32, 62, 112, 182, 252, 64, 126, 238, 420, 672, 924, 128, 254, 492, 912, 1584, 2508, 3432, PROG ;; PLT DrScheme from Joshua Zucker (define ht (make-hash-table 'equal)) (define (A039597 n k) (local ((define (help n k) (cond [(= n 0) (cond [(= k 0) 1] [else 0])] [(>= k n) 0] [else (hash-table-get ht (list n k) (lambda () (let ([answer (apply + (apply append (build-list n (lambda (n1) (build-list (cond [(= k 0) 1] [else k]) (lambda (k1) (help n1 k1)))))))]) (begin (hash-table-put! ht (list n k) answer) answer))))]))) (cond [(>= k n) 0] [else (apply + (apply append (build-list (add1 n) (lambda (n1) (build-list (add1 k) (lambda (k1) (help n1 k1)))))))]))) CROSSREFS See A274292 for another version of this triangle. Sequence in context: A162762 A156097 A288793 * A317979 A167229 A192333 Adjacent sequences:  A039594 A039595 A039596 * A039598 A039599 A039600 KEYWORD nonn,easy,tabl AUTHOR EXTENSIONS More terms from Joshua Zucker, Jun 22 2006 STATUS approved

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Last modified April 10 08:09 EDT 2021. Contains 342845 sequences. (Running on oeis4.)