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.
1, 1, 3, 3, 6, 5, 11, 8, 20, 14, 35, 24, 59, 41, 100, 77, 178, 162, 341, 364, 705, 837, 1542, 1915, 3458, 4282, 7741, 9280, 17021, 19461, 36482, 39559, 76042, 78218, 154261, 151184, 305445, 287509, 592954, 542223, 1135178, 1023210, 2158389, 1949312, 4107701
Offset: 1
Examples
For first seven rows of T see A131074 or A129961.
Links
- Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,22,-22,-24,24,20,-20,-10,10,4,-4).
Crossrefs
Programs
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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] ];
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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]), ", "))
Formula
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)).