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A101508 Product of binomial matrix and the Mobius matrix A051731.

%I A101508 #23 Aug 31 2024 08:33:42
%S A101508 1,2,1,4,2,1,8,4,3,1,16,8,6,4,1,32,16,11,10,5,1,64,32,21,20,15,6,1,
%T A101508 128,64,42,36,35,21,7,1,256,128,85,64,70,56,28,8,1,512,256,171,120,
%U A101508 127,126,84,36,9,1,1024,512,342,240,220,252,210,120,45,10,1,2048,1024,683,496,385,463,462,330,165,55,11,1
%N A101508 Product of binomial matrix and the Mobius matrix A051731.
%C A101508 Row sums are A101509. Diagonal sums are A101510.
%C A101508 The matrix inverse appears to be A128313. - _R. J. Mathar_, Mar 22 2013
%C A101508 Read as upper triangular matrix, this can be seen as "recurrences in A135356 applied to A023531" [_Paul Curtz_, Mar 03 2017]. - The columns are: A000079, A131577, A024495, A000749, A139761, ... Column n differs after the (n+1)-th nonzero term on from the binomial coefficients C(k,n). - _M. F. Hasler_, Mar 05 2017
%H A101508 Alois P. Heinz, <a href="/A101508/b101508.txt">Rows n = 0..140, flattened</a>
%F A101508 T(n, k) = Sum_{i=0..n} if(mod(i+1, k+1)=0, binomial(n, i), 0).
%F A101508 Rows have g.f. x^k/((1-x)^(k+1)-x^(k+1)).
%e A101508 Rows begin
%e A101508   1;
%e A101508   2,1;
%e A101508   4,2,1;
%e A101508   8,4,3,1;
%e A101508   16,8,6,4,1;
%e A101508   ...
%p A101508 A101508 := proc(n,k)
%p A101508     a := 0 ;
%p A101508     for i from 0 to n do
%p A101508         if modp(i+1,k+1) = 0 then
%p A101508             a := a+binomial(n,i) ;
%p A101508         end if;
%p A101508     end do:
%p A101508     return a;
%p A101508 end proc: # _R. J. Mathar_, Mar 22 2013
%t A101508 t[n_, k_] := Sum[If[Mod[i + 1, k + 1] == 0, Binomial[n, i], 0], {i, 0, n}]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 24 2014 *)
%o A101508 (PARI) T(n,k)=sum(i=0,n, if((i+1)%(k+1)==0, binomial(n, i))) \\ _M. F. Hasler_, Mar 05 2017
%K A101508 easy,nonn,tabl
%O A101508 0,2
%A A101508 _Paul Barry_, Dec 05 2004