A032443 -id:A032443 - cp's OEIS frontend

A111505 Right half of Pascal's triangle (A007318) with zeros.

1, 0, 1, 0, 2, 1, 0, 0, 3, 1, 0, 0, 6, 4, 1, 0, 0, 0, 10, 5, 1, 0, 0, 0, 20, 15, 6, 1, 0, 0, 0, 0, 35, 21, 7, 1, 0, 0, 0, 0, 70, 56, 28, 8, 1, 0, 0, 0, 0, 0, 126, 84, 36, 9, 1, 0, 0, 0, 0, 0, 252, 210, 120, 45, 10, 1, 0, 0, 0, 0, 0, 0, 462, 330, 165

Offset: 0

Philippe Deléham, Nov 16 2005

A034869 is the version without zeros.

			Triangle begins:
1;
0, 1;
0, 2, 1;
0, 0, 3, 1;
0, 0, 6, 4, 1;
0, 0, 0, 10, 5, 1;
0, 0, 0, 20, 15, 6, 1;
0, 0, 0, 0, 35, 21, 7, 1;
0, 0, 0, 0, 70, 56, 28, 8, 1;
0, 0, 0, 0, 0, 126, 84, 36, 9, 1;
0, 0, 0, 0, 0, 252, 210, 120, 45, 10, 1;
0, 0, 0, 0, 0, 0, 462, 330, 165, 55, 11, 1;
0, 0, 0, 0, 0, 0, 924, 792, 495, 220, 66, 12, 1;
0, 0, 0, 0, 0, 0, 0, 1716, 1287, 715, 286, 78, 13, 1;
0, 0, 0, 0, 0, 0, 0, 3432, 3003, 2002, 1001, 364, 91, 14, 1;

Cf. A007318, A034869, A094527, A111418.

Sum_{n, n>=k} T(n, k) = A001700(k).
Sum_{k =0..2*n} T(2*n, k) = A032443(n).
Sum_{k=0..2*n+1} T(2*n+1, k) = 4^n = A000302(n).
Sum_{k=0..2*n} T(2*n, k)^2 = A036910(n).
Sum_{k=0..2*n+1} T(2*n+1, k)^2 = C(4*n+1, 2*n) = A002458(n) . Paul D. Hanna

A367548 a(n) = Sum_{k = 0..n} binomial(-n, k) * 2^(n - k).

Original entry on oeis.org

1, 1, 3, -2, 19, -54, 222, -804, 3075, -11630, 44458, -170268, 654766, -2524508, 9758556, -37802952, 146724579, -570450078, 2221230066, -8660901612, 33811886394, -132148736148, 517012584036, -2024632609272, 7935337877454, -31126450260204, 122183595168612

Offset: 0

Peter Luschny, Nov 29 2023

sign

Cf. A032443.

seq(add(binomial(-n, k)*2^(n - k), k = 0..n), n = 0..26);

Table[Sum[Binomial[-n,k]2^(n-k),{k,0,n}],{n,0,30}] (* Harvey P. Dale, Apr 03 2024 *)

a(n) = 4^n*3^(-n) - binomial(-n, n+1) * hypergeom([1, 2*n+1], [n + 2], -1/2) / 2.
a(n) = [x^n] (3 + 12*x + sqrt(4*x + 1)*(4*x + 3))/(6 + 16*x - 32*x^2).
D-finite with recurrence 9*n*a(n) +6*(6*n-7)*a(n-1) +16*(-n-4)*a(n-2) +32*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jan 11 2024
From Seiichi Manyama, Jul 30 2025: (Start)
a(n) = [x^n] 1/((1-2*x) * (1+x)^n).
a(n) = Sum_{k=0..n} (-1)^k * 3^(n-k) * binomial(2*n,k). (End)

cp's OEIS Frontend

A111505 Right half of Pascal's triangle (A007318) with zeros.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A367548 a(n) = Sum_{k = 0..n} binomial(-n, k) * 2^(n - k).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula