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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
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jul 31 2011
EXTENSIONS
More terms from David A. Corneth, Oct 18 2016
STATUS
approved