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).

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

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

Table of n, a(n) for n=0..49.

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: A156097 A288793 A049015 * A317979 A359090 A167229

Adjacent sequences: A039594 A039595 A039596 * A039598 A039599 A039600

Keyword

nonn,easy,tabl

Author

N. J. A. Sloane.

Extensions

More terms from Joshua Zucker, Jun 22 2006

Status

approved