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A131077 Antidiagonal 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 A131077 #9 Jan 29 2025 08:07:56
%S A131077 1,1,3,3,6,5,11,8,20,14,35,24,59,41,100,77,178,162,341,364,705,837,
%T A131077 1542,1915,3458,4282,7741,9280,17021,19461,36482,39559,76042,78218,
%U A131077 154261,151184,305445,287509,592954,542223,1135178,1023210,2158389,1949312,4107701
%N A131077 Antidiagonal 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.
%H A131077 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-6,-15,15,22,-22,-24,24,20,-20,-10,10,4,-4).
%F A131077 G.f.: x*(1-4*x^2+6*x^4-x^5-4*x^6+3*x^7+x^8-3*x^9+x^10+2*x^11-x^12) / ((1-x)*(1-2*x^2)*(1+x^4)*(1-4*x^2+6*x^4-4*x^6+2*x^8)).
%e A131077 For first seven rows of T see A131074 or A129961.
%o A131077 (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+1)\2, M[j-k+1, k]), ", "))
%o A131077 (Magma) m:=44; 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+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ];
%Y A131077 Cf. A131074 (T read by rows), A129961 (main diagonal of T), A131075 (first subdiagonal of T), A131076 (row sums of T). First through sixth column of T are in A131078, A131079, A131080, A131081, A131082, A131083 resp.
%K A131077 nonn,easy
%O A131077 1,3
%A A131077 _Klaus Brockhaus_, following a suggestion of _Paul Curtz_, Jun 14 2007