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A222405 Triangle read by rows: left and right edges are A002061 (1,3,7,13,21,...), interior entries are filled in using the Pascal triangle rule.

%I A222405 #8 Jan 14 2014 10:03:36
%S A222405 1,3,3,7,6,7,13,13,13,13,21,26,26,26,21,31,47,52,52,47,31,43,78,99,
%T A222405 104,99,78,43,57,121,177,203,203,177,121,57,73,178,298,380,406,380,
%U A222405 298,178,73,91,251,476,678,786,786,678,476,251,91,111,342,727,1154,1464,1572,1464,1154,727,342,111
%N A222405 Triangle read by rows: left and right edges are A002061 (1,3,7,13,21,...), interior entries are filled in using the Pascal triangle rule.
%e A222405 Triangle begins:
%e A222405 1
%e A222405 3, 3
%e A222405 7, 6, 7
%e A222405 13, 13, 13, 13
%e A222405 21, 26, 26, 26, 21
%e A222405 31, 47, 52, 52, 47, 31
%e A222405 43, 78, 99, 104, 99, 78, 43
%e A222405 57, 121, 177, 203, 203, 177, 121, 57
%e A222405 73, 178, 298, 380, 406, 380, 298, 178, 73
%e A222405 ...
%p A222405 d:=[seq(n*(n+1)+1,n=0..14)];
%p A222405 f:=proc(d) local T,M,n,i;
%p A222405 M:=nops(d);
%p A222405 T:=Array(0..M-1,0..M-1);
%p A222405 for n from 0 to M-1 do T[n,0]:=d[n+1]; T[n,n]:=d[n+1]; od:
%p A222405 for n from 2 to M-1 do
%p A222405 for i from 1 to n-1 do T[n,i]:=T[n-1,i-1]+T[n-1,i]; od: od:
%p A222405 lprint("triangle:");
%p A222405 for n from 0 to M-1 do lprint(seq(T[n,i],i=0..n)); od:
%p A222405 lprint("row sums:");
%p A222405 lprint([seq( add(T[i,j],j=0..i), i=0..M-1)]);
%p A222405 end;
%p A222405 f(d);
%t A222405 t[n_, n_] := n^2+n+1; t[n_, 0] := n^2+n+1; t[n_, k_] := t[n, k] = t[n-1, k-1] + t[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 14 2014 *)
%Y A222405 Cf. A007318, A002061, A222403, A222404.
%Y A222405 Row sums are A027178.
%K A222405 nonn,tabl
%O A222405 0,2
%A A222405 _N. J. A. Sloane_, Feb 18 2013