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A102363 Triangle read by rows, constructed by a Pascal-like rule with left edge = 2^k, right edge = 2^(k+1)-1 (k >= 0).

%I A102363 #40 Dec 22 2018 03:32:23
%S A102363 1,2,3,4,5,7,8,9,12,15,16,17,21,27,31,32,33,38,48,58,63,64,65,71,86,
%T A102363 106,121,127,128,129,136,157,192,227,248,255,256,257,265,293,349,419,
%U A102363 475,503,511,512,513,522,558,642,768,894,978,1014,1023,1024,1025,1035,1080,1200,1410,1662,1872,1992,2037,2047
%N A102363 Triangle read by rows, constructed by a Pascal-like rule with left edge = 2^k, right edge = 2^(k+1)-1 (k >= 0).
%C A102363 First column right of center divided by 3 equals powers of 4.
%C A102363 Right of left edge, sums of rows are divisible by 3.
%C A102363 Apparently the number of terms per row plus the number of numbers in natural order skipped per row equals a power of 2. - _David Williams_, Jun 27 2009
%H A102363 Alois P. Heinz, <a href="/A102363/b102363.txt">Rows n = 0..140, flattened</a>
%F A102363 G.f.: Sum_{n>=0} x^n * (1+x)^tr(n) = Sum_{n>=0} a(n)*x^n, where tr(n) = A002024(n+1) = floor(sqrt(2*n+1) + 1/2). - _Paul D. Hanna_, Feb 19 2016
%F A102363 G.f.: Sum_{n>=1} x^(n*(n-1)/2) * (1-x^n)/(1-x) * (1+x)^n = Sum_{n>=0} a(n)*x^n. - _Paul D. Hanna_, Feb 20 2016
%F A102363 a(n) = A007318(n-1) + a(n-1). - _Jon Maiga_, Dec 22 2018
%e A102363 This triangle begins:
%e A102363                             1
%e A102363                          2     3
%e A102363                       4     5     7
%e A102363                    8     9    12    15
%e A102363                16    17    21    27    31
%e A102363             32    33    38    48    58    63
%e A102363          64    65    71    86   106   121   127
%e A102363      128   129   136   157   192   227   248   255
%e A102363   256   257   265   293   349   419   475   503   511
%e A102363 G.f. of this sequence in flattened form:
%e A102363 A(x) = 1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 7*x^5 + 8*x^6 + 9*x^7 + 12*x^8 + 15*x^9 + 16*x^10 + 17*x^11 + 21*x^12 + 27*x^13 + 31*x^14 + 32*x^15 + ...
%e A102363 such that
%e A102363 A(x) = (1+x) + x*(1+x)^2 + x^2*(1+x)^2 + x^3*(1+x)^3 + x^4*(1+x)^3 + x^5*(1+x)^3 + x^6*(1+x)^4 + x^7*(1+x)^4 + x^8*(1+x)^4 + x^9*(1+x)^4 + x^10*(1+x)^5 + x^11*(1+x)^5 + x^12*(1+x)^5 + x^13*(1+x)^5 + x^14*(1+x)^5 + x^15*(1+x)^6 + ...
%p A102363 T:=proc(n,k) if k=0 then 2^n elif k=n then 2^(n+1)-1 else T(n-1,k)+T(n-1,k-1) fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form - _Emeric Deutsch_, Mar 26 2005
%t A102363 t[n_, 0] := 2^n; t[n_, n_] := 2^(n+1)-1; t[n_, k_] := t[n, k] = t[n-1, k] + t[n-1, k-1]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 15 2013 *)
%o A102363 (PARI) /* Print in flattened form: Sum_{n>=0} x^n*(1+x)^tr(n) */
%o A102363 {tr(n) = ceil( (sqrt(8*n+9)-1)/2 )}
%o A102363 {a(n) = polcoeff( sum(m=0,n, x^m * (1+x +x*O(x^n))^tr(m) ),n)}
%o A102363 for(n=0,78, print1(a(n),", ")) \\ _Paul D. Hanna_, Feb 19 2016
%Y A102363 Cf. A000079, A053220 (row sums), A265939 (central terms).
%K A102363 nonn,tabl,easy
%O A102363 0,2
%A A102363 _David Williams_, Mar 15 2005, Oct 05 2007
%E A102363 More terms from _Emeric Deutsch_, Mar 26 2005