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A131076 Row sums of triangular array T: T(j,1) = 1 for ((j-1) mod 8) < 4, else 0; T(j,k) = T(j-1,k-1) + T(j,k-1) for 2 <= k <= j.

%I A131076 #11 Jun 23 2025 13:22:43
%S A131076 1,3,7,15,26,42,64,93,139,231,463,1092,2744,6840,16384,37383,81295,
%T A131076 169119,338239,654192,1232288,2280864,4194304,7761375,14635711,
%U A131076 28384383,56768767,116566080,243472256,511907712,1073741824,2232713343,4585959679,9292452351
%N A131076 Row sums of triangular array T: T(j,1) = 1 for ((j-1) mod 8) < 4, else 0; T(j,k) = T(j-1,k-1) + T(j,k-1) for 2 <= k <= j.
%C A131076 Sum of n-th row equals (n+1)-th term of main diagonal minus (n+1)-th term of first column: a(n) = A129961(n+1) - A131078(n+1).
%H A131076 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (7,-20,30,-27,21,-24,30,-26,14,-4).
%F A131076 G.f.: x*(1-4*x+6*x^2-4*x^3-2*x^4+10*x^5-10*x^6+5*x^7-x^8)/((1-x)*(1-2*x)*(1+x^4)*(1-4*x+6*x^2-4*x^3+2*x^4)).
%e A131076 For first seven rows of T see A131074 or A129961.
%t A131076 LinearRecurrence[{7,-20,30,-27,21,-24,30,-26,14,-4},{1,3,7,15,26,42,64,93,139,231},40] (* _Harvey P. Dale_, Jun 23 2025 *)
%o A131076 (Magma) m:=32; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 8 lt 4 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; [ &+[ M[j, k]: k in [1..j] ]: j in [1..m] ];
%o A131076 (PARI) lista(m) = my(M=matrix(m, m)); for(j=1, m, M[j, 1]=if((j-1)%8<4, 1, 0)); for(k=2, m, for(j=k, m, M[j, k]=M[j-1, k-1]+M[j, k-1])); for(j=1, m, print1(sum(k=1, j, M[j, k]), ", "))
%Y A131076 Cf. A131074 (T read by rows), A129961 (main diagonal of T), A131075 (first subdiagonal of T), A131077 (antidiagonal sums of T). First through sixth column of T are in A131078, A131079, A131080, A131081, A131082, A131083 resp.
%K A131076 nonn,easy
%O A131076 1,2
%A A131076 _Klaus Brockhaus_, following a suggestion of _Paul Curtz_, Jun 14 2007