A259525 First differences of A007318, when Pascal's triangle is seen as flattened list.

0, 0, 0, 1, -1, 0, 2, 0, -2, 0, 3, 2, -2, -3, 0, 4, 5, 0, -5, -4, 0, 5, 9, 5, -5, -9, -5, 0, 6, 14, 14, 0, -14, -14, -6, 0, 7, 20, 28, 14, -14, -28, -20, -7, 0, 8, 27, 48, 42, 0, -42, -48, -27, -8, 0, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9, 0, 10, 44, 110

Offset: 0

Author

Reinhard Zumkeller, Jul 18 2015

Keywords

• sign

Comments

A214292 gives first differences per row in Pascal's triangle.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A007318, A014473, A163866, A214292.

Cf. A000108, A021499, A047171, A074331.

Programs

Haskell

a259525 n = a259525_list !! n
a259525_list = zipWith (-) (tail pascal) pascal
                           where pascal = concat a007318_tabl

Magma

[k eq n select 0 else (n-2*k-1)*Binomial(n,k+1)/(n-k): k in [0..n], n in [0..14]]; // G. C. Greubel, Apr 25 2024

Mathematica

Table[If[k==n, 0, ((n-2*k-1)/(n-k))*Binomial[n,k+1]], {n,0,12}, {k,0, n}]//Flatten (* G. C. Greubel, Apr 25 2024 *)

SageMath

flatten([[binomial(n,k+1) -binomial(n,k) +int(k==n) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Apr 25 2024

Formula

From G. C. Greubel, Apr 25 2024: (Start)

If viewed as a triangle then:

T(n, k) = binomial(n, k+1) - binomial(n, k), with T(n, n) = 0.

T(n, n-k) = - T(n, k), for 0 <= k < n.

T(2*n, n) = [n=0] - A000108(n).

Sum_{k=0..n} T(n, k) = 0 (row sums).

Sum_{k=0..floor(n/2)} T(n, k) = A047171(n).

Sum_{k=0..n} (-1)^k*T(n, k) = A021499(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A074331(n-1). (End)