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 A193605 Triangle: (row n) = partial sums of partial sums of row n of Pascal's triangle. 3
 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, 192, 256, 1, 9, 38, 102, 201, 321, 448, 576, 1, 10, 47, 140, 303, 522, 769, 1024, 1280, 1, 11, 57, 187, 443, 825, 1291, 1793, 2304, 2816, 1, 12, 68, 244, 630, 1268, 2116, 3084, 4097, 5120, 6144 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 LINKS Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3. FORMULA Writing the general term as T(n,k), for 0<=k<=n: T(n,n)=A001792, T(n,n-1)=A001787, T(n,n-2)=A000337, T(n,n-3)=A045618. T(n-1,k-1) + T(n-1,k) = T(n,k). - David A. Corneth, Oct 18 2016 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 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 EXAMPLE First 5 rows of A193605: 1 1....3 1....4....8 1....5....12....20 1....6....17....32....48 MAPLE A052509 := proc(n, k)     if k = 0 then         1;     else         procname(n, k-1)+binomial(n, k) ;     end if; end proc: A193605 := proc(n, k)     if k = 0 then         1;     else         procname(n, k-1)+A052509(n, k) ;     end if; end proc: # R. J. Mathar, Apr 22 2013 # Alternative after Vladimir Kruchinin: gf := ((x*y-1)/(1-2*x*y))^2/(1-x*y-x): ser := series(gf, x, 12): p := n -> coeff(ser, x, n): row := n -> seq(coeff(p(n), y, k), k=0..n): seq(row(n), n=0..10); # Peter Luschny, Aug 19 2019 MATHEMATICA u[n_, k_] := Sum[Binomial[n, h], {h, 0, k}] p[n_, k_] := Sum[u[n, h], {h, 0, k}] Table[p[n, k], {n, 0, 12}, {k, 0, n}] Flatten[%]   (* A193605 as a sequence *) TableForm[Table[p[n, k], {n, 0, 12}, {k, 0, n}]]  (* A193605 as a triangle *) PROG (Maxima) T(n, k):=sum(((i+3)*2^(i-2))*binomial(n-i, k-i), i, 1, min(n, k))+binomial(n, k); /* Vladimir Kruchinin, Aug 20 2019 */ CROSSREFS Cf. A193606. Sequence in context: A081255 A005371 A210739 * A193667 A205878 A329130 Adjacent sequences:  A193602 A193603 A193604 * A193606 A193607 A193608 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Jul 31 2011 EXTENSIONS More terms from David A. Corneth, Oct 18 2016 STATUS approved

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Last modified May 31 00:29 EDT 2020. Contains 334747 sequences. (Running on oeis4.)