A131026 -id:A131026 - cp's OEIS frontend

A130869 Partial sums of A130752.

2, 7, 16, 32, 63, 126, 254, 511, 1024, 2048, 4095, 8190, 16382, 32767, 65536, 131072, 262143, 524286, 1048574, 2097151, 4194304, 8388608, 16777215, 33554430, 67108862, 134217727, 268435456, 536870912, 1073741823, 2147483646

Offset: 0

Paul Curtz, Jul 24 2007

nonn

O.g.f.: (2-x)/[(1-x)(1-2x)(1-x+x^2)]. - R. J. Mathar, Jul 04 2008
Inverse binomial transform yields 2,(5,4,3) periodically continued with period 3. - R. J. Mathar, Jul 04 2008
a(n+1)-2a(n) = A131290(n+1). 2^(n+2)-a(n) = A131026(n+2).

Edited and extended by R. J. Mathar, Jul 04 2008

A177767 Triangle read by rows: T(n,k) = binomial(n - 1, k - 1), 1 <= k <= n, and T(n,0) = A153881(n+1), n >= 0.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, May 13 2010

Row sums yield A000225 preceded by 1.

Except for signs, this is A135225.

			Triangle begins:
   1;
  -1, 1;
  -1, 1, 1;
  -1, 1, 2,  1;
  -1, 1, 3,  3,  1;
  -1, 1, 4,  6,  4,   1;
  -1, 1, 5, 10, 10,   5,   1;
  -1, 1, 6, 15, 20,  15,   6,  1;
  -1, 1, 7, 21, 35,  35,  21,  7,  1;
  -1, 1, 8, 28, 56,  70,  56, 28,  8, 1;
  -1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;
   ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000225, A000071, A007318, A014473, A097805, A131026, A135225.

A177767:= func< n,k | k eq n select 1 else  k eq 0 select -1 else Binomial(n-1, k-1) >;
[A177767(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Apr 22 2024

Flatten[Table[If[n == 0, {1}, CoefficientList[x*(1 + x)^( n - 1) - 1, x]], {n, 0, 10}]]

T(n, k) := if k = 0 then 2*floor(1/(n + 1)) - 1 else binomial(n - 1, k - 1)$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Oct 23 2018 */

flatten([[binomial(n-1, k-1) - int(k==0) + 2*int(n==0) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 22 2024

Row n = coefficients in the expansion of x*(1 + x)^(n - 1) - 1, n > 0.
From Franck Maminirina Ramaharo, Oct 23 2018: (Start)
G.f.: (1 - 3*y + (2 + x)*y^2)/(1 - (2 + x)*y + (1 + x)*y^2).
E.g.f.: (2 + x - (1 + x)*exp(y) + x*exp((1 + x)*y))/(1 + x). (End)
From G. C. Greubel, Apr 22 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A153881(n+1) - [n=1].
Sum_{k=0..floor(n/2)} T(n-k, k) = A000071(n-1) + [n=0].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = -A131026(n-1) + [n=0]. (End)

Edited and new name by Franck Maminirina Ramaharo, Oct 23 2018

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A130869 Partial sums of A130752.

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A177767 Triangle read by rows: T(n,k) = binomial(n - 1, k - 1), 1 <= k <= n, and T(n,0) = A153881(n+1), n >= 0.

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