A366936 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^j*j^k*floor(n/j).

-1, -1, -1, -1, 0, -3, -1, 2, -4, -2, -1, 6, -8, 1, -4, -1, 14, -22, 11, -5, -4, -1, 30, -68, 49, -15, -1, -6, -1, 62, -214, 203, -77, 15, -9, -4, -1, 126, -668, 841, -423, 119, -35, 4, -7, -1, 254, -2062, 3491, -2285, 807, -225, 48, -9, -7, -1, 510, -6308, 14449

Offset

1,6

Links

Table of n, a(n) for n=1..59.

Formula

Let A(n, k) = Sum_{j=1..n} j^k * floor(n/j). Then T(n, k) = 2^(k+1)*A(floor(n/2), k) - A(n, k).

Example

Array begins:
-1, -1,  -1,  -1,   -1,    -1,     -1,     -1,      -1,       -1, ...
-1,  0,   2,   6,   14,    30,     62,    126,     254,      510, ...
-3, -4,  -8, -22,  -68,  -214,   -668,  -2062,   -6308,   -19174, ...
-2,  1,  11,  49,  203,   841,   3491,  14449,   59483,   243481, ...
-4, -5, -15, -77, -423, -2285, -12135, -63677, -331143, -1709645, ...

Prog

(Python)
from math import isqrt
from itertools import count, islice
from sympy import bernoulli
def A366936_T(n, k):
    if k:
        return ((((s:=isqrt(m:=n>>1))+1)*(bernoulli(k+1)-bernoulli(k+1, s+1))<<k+1)-((t:=isqrt(n))+1)*(bernoulli(k+1)-bernoulli(k+1, t+1))+(sum(w**k*(k+1)*((q:=m//w)+1)-bernoulli(k+1)+bernoulli(k+1, q+1) for w in range(1, s+1))<<k+1)-sum(w**k*(k+1)*((q:=n//w)+1)-bernoulli(k+1)+bernoulli(k+1, q+1) for w in range(1, t+1)))//(k+1) if n else -1
    else:
        return (s:=isqrt(n))**2-((t:=isqrt(m:=n>>1))**2<<1)+((sum(m//k for k in range(1, t+1))<<1)-sum(n//k for k in range(1, s+1))<<1)
def A366936_gen(): return (A366936_T(k+1, n-k-1) for n in count(1) for k in range(n))
A366936_list = list(islice(A366936_gen(), 30))

Crossrefs

First column is -A059851.

Second column is A024919.

Third column is A366915.

Fourth column is A366917.

First row is -A000012.

Second row is A000918.

First superdiagonal is A366919.

Cf. A319649.

Sequence in context: A280348 A332397 A004608 * A201673 A343764 A117905

Adjacent sequences: A366933 A366934 A366935 * A366937 A366938 A366939

Keyword

sign,tabl

Author

Chai Wah Wu, Oct 29 2023

Status

approved