A254876 -id:A254876 - cp's OEIS frontend

A088488 a(n) = Sum_{k=1..8} floor(A254876(n,k)/A254876(n-1,k)), where A254876(n,k) = n! / (Product_{m=(n-floor((2n)/(3^k))) .. (n-floor((n)/(3^k)))} m).

8, 17, 22, 26, 40, 43, 49, 66, 65, 69, 87, 87, 68, 109, 108, 109, 137, 130, 130, 157, 153, 133, 180, 174, 171, 211, 196, 191, 227, 218, 186, 250, 240, 232, 280, 262, 253, 298, 285, 164, 319, 304, 292, 350, 327, 313, 367, 349, 292, 390, 371, 354, 426, 393, 375

Offset: 2

Roger L. Bagula, Nov 09 2003

Antti Karttunen, Table of n, a(n) for n = 2..10000

Cf. A088487, A254876.

p[n_, k_]=n!/Product[i, {i, n-Floor[2*n/3^k], n-Floor[n/3^k]}]
f[n_]=Sum[Floor[p[n, k]/p[n-1, k]], {k, 1, 8}]
at=Table[f[n], {n, 2, 200}]

A088488(n) = sum(k=1,8,(A254876bi(n,k)\A254876bi(n-1,k)));
A254876bi(n,k) = n! / prod(i=(n-((2*n)\(3^k))),(n-(n\(3^k))),i);
for(n=2, 10000, write("b088488.txt", n, " ", A088488(n)));

(define (A088488 n) (add (lambda (k) (floor->exact (/ (A254876bi n k) (A254876bi (- n 1) k)))) 1 8))
;; The following function implements sum_{i=lowlim..uplim} intfun(i)
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{k=1..8} floor(A254876(n,k)/A254876(n-1,k)), where A254876(n,k) = n! / (Product_{m=(n-floor((2n)/(3^k))) .. (n-floor((n)/(3^k)))} m).

Edited by Antti Karttunen, Feb 09 2015

A254864 Triangular table T(n,k) = n! / (n-floor(n/3^k))!, read by rows T(1,1), T(2,1), T(2,2), T(3,1), T(3,2), T(3,3), ...

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 1, 1, 1, 5, 1, 1, 1, 1, 30, 1, 1, 1, 1, 1, 42, 1, 1, 1, 1, 1, 1, 56, 1, 1, 1, 1, 1, 1, 1, 504, 9, 1, 1, 1, 1, 1, 1, 1, 720, 10, 1, 1, 1, 1, 1, 1, 1, 1, 990, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11880, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17160, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24024, 14, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Feb 09 2015

An auxiliary array for computing A088487.

			The first 27 rows of a triangular table:
1
1, 1
3, 1, 1
4, 1, 1, 1
5, 1, 1, 1, 1
30, 1, 1, 1, 1, 1
42, 1, 1, 1, 1, 1, 1
56, 1, 1, 1, 1, 1, 1, 1
504, 9, 1, 1, 1, 1, 1, 1, 1
720, 10, 1, 1, 1, 1, 1, 1, 1, 1
990, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1
11880, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
17160, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
24024, 14, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
360360, 15, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
524160, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
742560, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
13366080, 306, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
19535040, 342, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
27907200, 380, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
586051200, 420, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
859541760, 462, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
1235591280, 506, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
29654190720, 552, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
43609104000, 600, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
62990928000, 650, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1700755056000, 17550, 27, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
...
(the last ones truncated a bit).

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 rows of triangular table

The leftmost column: A254865.

Cf. A000142, A000244, A088487, A254876.

A254864bi(n,k) = prod(i=(1+(n-(n\(3^k)))),n,i);

(define (A254864 n) (A254864bi (A002024 n) (A002260 n)))
;; The above function can then use either one of these:
(define (A254864bi n k) (/ (A000142 n) (A000142 (- n (floor->exact (/ n (expt 3 k)))))))
(define (A254864bi n k) (mul A000027 (+ 1 (- n (floor->exact (/ n (expt 3 k))))) n))
(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))

T(n,k) = n! / (n-floor(n/3^k))! = A000142(n) / A000142(n-floor(n/A000244(k))).

T(n,k) = Product_{m=1+(n-floor(n/(3^k))) .. n} m.

cp's OEIS Frontend

A088488 a(n) = Sum_{k=1..8} floor(A254876(n,k)/A254876(n-1,k)), where A254876(n,k) = n! / (Product_{m=(n-floor((2n)/(3^k))) .. (n-floor((n)/(3^k)))} m).

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