%I A039597
%S A039597 2,4,6,8,14,20,16,30,50,70,32,62,112,182,252,64,126,238,420,672,924,128,
%T A039597 254,492,912,1584,2508,3432,256,510,1002,1914,3498,6006,9438,12870,512,
%U A039597 1022,2024,3938,7436,13442,22880,35750,48620,1024,2046,4070,8008,15444
%N A039597 Triangle a(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 A039597 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 A039597 Carlitz gives recurrence.
%e A039597 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).
%o A039597 ;;PLT DrScheme from Joshua Zucker
%o A039597 (define ht (make-hash-table 'equal))
%o A039597 (define (A039597 n k)
%o A039597 (local ((define (help n k)
%o A039597 (cond
%o A039597 [(= n 0) (cond [(= k 0) 1] [else 0])]
%o A039597 [(>= k n) 0]
%o A039597 [else
%o A039597 (hash-table-get ht (list n k) (lambda ()
%o A039597 (let ([answer (apply + (apply append (build-list n (lambda (n1) (build-list 
               (cond [(= k 0) 1] [else k])
%o A039597 (lambda (k1) (help n1 k1)))))))])
%o A039597 (begin (hash-table-put! ht (list n k) answer)
%o A039597 answer))))])))
%o A039597 (cond
%o A039597 [(>= k n) 0]
%o A039597 [else (apply + (apply append (build-list (add1 n) (lambda (n1) (build-list 
               (add1 k) (lambda (k1) (help n1 k1)))))))])))
%Y A039597 Sequence in context: A005250 A162762 A156097 this_sequence A000937 A167229 
               A068902
%Y A039597 Adjacent sequences: A039594 A039595 A039596 this_sequence A039598 A039599 
               A039600
%K A039597 nonn,easy,tabl
%O A039597 0,1
%A A039597 N. J. A. Sloane (njas(AT)research.att.com).
%E A039597 More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), 
               Jun 22 2006