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

0,3

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

Table of n, a(n) for n=0..65.

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