A172092 - OEIS

%I #7 Sep 08 2022 08:45:50

%S 1,1,1,1,-2,1,1,-47,-47,1,1,-2027,-2072,-2027,1,1,-249599,-251624,

%T -251624,-249599,1,1,-91359839,-91609436,-91611416,-91609436,

%U -91359839,1,1,-100039779839,-100131139676,-100131389228,-100131389228,-100131139676,-100039779839,1

%N Triangle, read by rows, T(n,k,q) = c(k,q) + c(n-k,q) - c(n, q) where c(n,q) = Product_{j=1..n-1} ((q^(j+1) - 1)/(q-1)) and q = 3.

%C Row sums are: {1, 2, 0, -92, -6124, -1002444, -457549964, -600604617484, -2298816299112204, -25856055844713627404, -858811326017167374184204, ...}.

%H G. C. Greubel, <a href="/A172092/b172092.txt">Rows n = 0..50 of triangle, flattened</a>

%F Let c(n,q) = Product_{j=1..n-1} ((q^(j+1) - 1)/(q-1)) then T(n,k,q) = -c(n,q) + c(n-k,q) + c(k, q) for q=3.

%e Triangle begins as:

%e   1;

%e   1,         1;

%e   1,        -2,         1;

%e   1,       -47,       -47,         1;

%e   1,     -2027,     -2072,     -2027,         1;

%e   1,   -249599,   -251624,   -251624,   -249599,         1;

%e   1, -91359839, -91609436, -91611416, -91609436, -91359839, 1;

%p T:= proc(n, k, q) option remember;

%p     c(n,q):= mul( (q^(j+1) -1)/(q-1), j=1..n-1);

%p     T(n,k,q):= c(k,q) + c(n-k,q) - c(n,q);

%p    end:

%p seq(seq(T(n,k,3), k=0..n), n=0..10); # _G. C. Greubel_, Dec 05 2019

%t c[n_, q_]:= Product[(q^(j+1) -1)/(q-1), {j, n-1}]; T[n_, k_, q_]:= c[k, q] + c[n-k, q] - c[n, q]; Table[T[n, k, 3], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Dec 05 2019 *)

%o (PARI) c(n,q) = prod(j=1, n-1, (q^(j+1) -1)/(q-1));

%o T(n, k, q) = c(k,q) + c(n-k,q) - c(n,q);

%o for(n=0, 10, for(k=0, n, print1(T(n,k,3), ", "))) \\ _G. C. Greubel_, Dec 05 2019

%o (Magma) c:= func< n,q | n lt 2 select 1 else &*[(q^(j+1) -1)/(q-1): j in [1..n-1]] >;

%o T:= func< n,k,q | c(k,q) + c(n-k,q) - c(n,q) >;

%o [T(n,k,3): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 05 2019

%o (Sage)

%o def c(n,q): return product( (q^(j+1) -1)/(q-1) for j in (1..n-1))

%o def T(n,k,q): return c(k,q) + c(n-k,q) - c(n,q)

%o [[T(n,k,3) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 05 2019

%Y Cf. A172091 (q=2), this sequence (q=3), A172093 (q=4).

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, Jan 25 2010

%E Edited by _G. C. Greubel_, Dec 05 2019