A193605 Triangle: (row n) = partial sums of partial sums of row n of Pascal's triangle.
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
Offset: 0
Examples
First 5 rows of A193605: 1 1....3 1....4....8 1....5....12....20 1....6....17....32....48
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.
Crossrefs
Cf. A193606.
Programs
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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
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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 *) -
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 */
Formula
Writing the general term as T(n,k), for 0<=k<=n:
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
Extensions
More terms from David A. Corneth, Oct 18 2016
Comments