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A193605 Triangle: (row n) = partial sums of partial sums of row n of Pascal's triangle.

%I A193605 #27 Aug 20 2019 02:16:27
%S A193605 1,1,3,1,4,8,1,5,12,20,1,6,17,32,48,1,7,23,49,80,112,1,8,30,72,129,
%T A193605 192,256,1,9,38,102,201,321,448,576,1,10,47,140,303,522,769,1024,1280,
%U A193605 1,11,57,187,443,825,1291,1793,2304,2816,1,12,68,244,630,1268,2116,3084,4097,5120,6144
%N A193605 Triangle: (row n) = partial sums of partial sums of row n of Pascal's triangle.
%C A193605 The n-th row is contains the partial sums of the n-th row of the array interpretation of A052509. - _R. J. Mathar_, Apr 22 2013
%H A193605 Denis Neiter and Amsha Proag, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Proag/proag3.html">Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
%F A193605 Writing the general term as T(n,k), for 0<=k<=n:
%F A193605 T(n,n)=A001792, T(n,n-1)=A001787, T(n,n-2)=A000337, T(n,n-3)=A045618.
%F A193605 T(n-1,k-1) + T(n-1,k) = T(n,k). - _David A. Corneth_, Oct 18 2016
%F A193605 G.f.: -(1-x*y)^2/(4*x^3*y^3+(4*x^3-8*x^2)*y^2+(5*x-4*x^2)*y+x-1). - _Vladimir Kruchinin_, Aug 19 2019
%F A193605 T(n,k) = C(n,k)+Sum_{i=1..n} (i+3)*2^(i-2)*C(n-i,k-i), - _Vladimir Kruchinin_, Aug 20 2019
%e A193605 First 5 rows of A193605:
%e A193605 1
%e A193605 1....3
%e A193605 1....4....8
%e A193605 1....5....12....20
%e A193605 1....6....17....32....48
%p A193605 A052509 := proc(n,k)
%p A193605     if k = 0 then
%p A193605         1;
%p A193605     else
%p A193605         procname(n,k-1)+binomial(n,k) ;
%p A193605     end if;
%p A193605 end proc:
%p A193605 A193605 := proc(n,k)
%p A193605     if k = 0 then
%p A193605         1;
%p A193605     else
%p A193605         procname(n,k-1)+A052509(n,k) ;
%p A193605     end if;
%p A193605 end proc: # _R. J. Mathar_, Apr 22 2013
%p A193605 # Alternative after _Vladimir Kruchinin_:
%p A193605 gf := ((x*y-1)/(1-2*x*y))^2/(1-x*y-x): ser := series(gf, x, 12):
%p A193605 p := n -> coeff(ser,x,n): row := n -> seq(coeff(p(n),y,k), k=0..n):
%p A193605 seq(row(n), n=0..10); # _Peter Luschny_, Aug 19 2019
%t A193605 u[n_, k_] := Sum[Binomial[n, h], {h, 0, k}]
%t A193605 p[n_, k_] := Sum[u[n, h], {h, 0, k}]
%t A193605 Table[p[n, k], {n, 0, 12}, {k, 0, n}]
%t A193605 Flatten[%]   (* A193605 as a sequence *)
%t A193605 TableForm[Table[p[n, k], {n, 0, 12}, {k, 0, n}]]  (* A193605 as a triangle *)
%o A193605 (Maxima)
%o A193605 T(n,k):=sum(((i+3)*2^(i-2))*binomial(n-i,k-i),i,1,min(n,k))+binomial(n,k);
%o A193605 /* _Vladimir Kruchinin_, Aug 20 2019 */
%Y A193605 Cf. A193606.
%K A193605 nonn,tabl
%O A193605 0,3
%A A193605 _Clark Kimberling_, Jul 31 2011
%E A193605 More terms from _David A. Corneth_, Oct 18 2016