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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

%I #14 Jun 18 2016 00:13:19

%S 2,4,6,8,14,20,16,30,50,70,32,62,112,182,252,64,126,238,420,672,924,

%T 128,254,492,912,1584,2508,3432,256,510,1002,1914,3498,6006,9438,

%U 12870,512,1022,2024,3938,7436,13442,22880,35750,48620,1024,2046,4070,8008,15444

%N 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).

%D 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.

%F 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

%e 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).

%e Triangle begins:

%e 2,

%e 4, 6,

%e 8, 14, 20,

%e 16, 30, 50, 70,

%e 32, 62, 112, 182, 252,

%e 64, 126, 238, 420, 672, 924,

%e 128, 254, 492, 912, 1584, 2508, 3432,

%o ;;PLT DrScheme from Joshua Zucker

%o (define ht (make-hash-table 'equal))

%o (define (A039597 n k)

%o (local ((define (help n k)

%o (cond

%o [(= n 0) (cond [(= k 0) 1] [else 0])]

%o [(>= k n) 0]

%o [else

%o (hash-table-get ht (list n k) (lambda ()

%o (let ([answer (apply + (apply append (build-list n (lambda (n1) (build-list (cond [(= k 0) 1] [else k])

%o (lambda (k1) (help n1 k1)))))))])

%o (begin (hash-table-put! ht (list n k) answer)

%o answer))))])))

%o (cond

%o [(>= k n) 0]

%o [else (apply + (apply append (build-list (add1 n) (lambda (n1) (build-list (add1 k) (lambda (k1) (help n1 k1)))))))])))

%Y See A274292 for another version of this triangle.

%K nonn,easy,tabl

%O 0,1

%A _N. J. A. Sloane_.

%E More terms from _Joshua Zucker_, Jun 22 2006