A099570 -id:A099570 - cp's OEIS frontend

A097141 Expansion of x(1+2x)/(1+x)^2.

0, 1, 0, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 34, -35, 36, -37, 38, -39, 40, -41, 42, -43, 44, -45, 46, -47, 48, -49, 50, -51, 52, -53, 54, -55, 56, -57, 58, -59, 60

Offset: 0

Paul Barry, Jul 29 2004

Partial sums of A097140.
Binomial transform is x(1+x)/(1-x), or {0,1,2,2,2,2,....}.
Second binomial transform is x/((1-x)^2(1 - 2x)), or Eulerian numbers A000295(n+1).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Cf. A000295, A038608, A040000, A097140, A099570.

[0] cat [(n-2)*(-1)^n : n in [1..100]]; // Wesley Ivan Hurt, Dec 11 2016

A097141:=n->(n-2)*(-1)^n: 0, seq(A097141(n), n=1..100); # Wesley Ivan Hurt, Dec 11 2016

CoefficientList[Series[x (1 + 2 x)/(1 + x)^2, {x, 0, 100}], x] (* Vincenzo Librandi, Mar 11 2014 *)

a(n)=if(n, (n-2)*(-1)^n, 0) \\ Charles R Greathouse IV, Dec 13 2016

G.f.: x*(1+2*x)/(1+x)^2.
a(n) = (n-2)*(-1)^n + 2*0^n.
a(n) = -2*a(n-1) - a(n-2) for n > 2.
a(n) = A099570(n) for n > 1. - R. J. Mathar, Dec 15 2008
a(n) = (Sum_{k=1..n} k*(-1)^(n-k)*binomial(n-1,k-1)*binomial(2*n-k-1,n-1))/n, n>0, a(0)=0. - Vladimir Kruchinin, Mar 09 2014
a(n) = A038608(n-2) for n > 2. - Georg Fischer, Oct 06 2018
E.g.f.: 2 - exp(-x)*(2 + x). - Stefano Spezia, Mar 07 2023

A099569 Riordan array (((1+x)^2 - x^3)/(1+x)^3, 1/(1+x)).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -2, 3, -3, 1, 4, -5, 6, -4, 1, -7, 9, -11, 10, -5, 1, 11, -16, 20, -21, 15, -6, 1, -16, 27, -36, 41, -36, 21, -7, 1, 22, -43, 63, -77, 77, -57, 28, -8, 1, -29, 65, -106, 140, -154, 134, -85, 36, -9, 1, 37, -94, 171, -246, 294, -288, 219, -121, 45, -10, 1, -46, 131, -265, 417, -540, 582, -507, 340, -166, 55, -11, 1

Offset: 0

Paul Barry, Oct 22 2004

Inverse matrix of A099567. Row sums are A099570.

			Rows begin as:
    1;
   -1,   1;
    1,  -2,    1;
   -2,   3,   -3,   1;
    4,  -5,    6,  -4,    1;
   -7,   9,  -11,  10,   -5,   1;
   11, -16,   20, -21,   15,  -6,   1;
  -16,  27,  -36,  41,  -36,  21,  -7,  1;
   22, -43,   63, -77,   77, -57,  28, -8,  1;
  -29,  65, -106, 140, -154, 134, -85, 36, -9,  1;

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

Cf. A076540, A099570 (row sums), A099567.

[n eq 0 select 1 else (-1)^(n+k)*(Binomial(n, k) + Binomial(n-1, k+2)): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 25 2022

C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if;
end proc:
for n from 0 to 10 do
    seq((-1)^(n+k)*(C(n, n-k) + add((i-2)*C(n-i, n-k-i), i = 3..n)), k = 0..n);
end do; # Peter Bala, Mar 21 2018

T[n_, k_]:= (-1)^(n+k)*(Binomial[n, k] + Binomial[n-1, k+2]);
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 25 2022 *)

def A099569(n, k): return 1 if (n==0) else (-1)^(n+k)*(binomial(n, k) +binomial(n-1, k+2))
flatten([[A099569(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 25 2022

Sum_{k=0..n} T(n, k) = A099570(n).
Columns have g.f. ((1+x)^2 - x^3)/(1+x)^3*(x/(1+x))^k.
T(n,k) = (-1)^(n+k)*(binomial(n, n-k) + Sum_{i = 3..n} (i-2)*binomial(n-i,n-k-i)), for 0 <= k <= n, otherwise 0. - Peter Bala, Mar 21 2018
From G. C. Greubel, Jul 25 2022: (Start)
T(n, k) = (-1)^(n+k)*(binomial(n, k) + binomial(n-1, k+2)), with T(0, k) = 1.
T(2*n-1, n-1) = (-1)^n*A076540(n), n >= 1.
T(n, n-1) = -n. (End)

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A097141 Expansion of x(1+2x)/(1+x)^2.

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A099569 Riordan array (((1+x)^2 - x^3)/(1+x)^3, 1/(1+x)).

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cp's OEIS Frontend

A097141 Expansion of x*(1+2*x)/(1+x)^2.

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Magma

Maple

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A099569 Riordan array (((1+x)^2 - x^3)/(1+x)^3, 1/(1+x)).

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Magma

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A097141 Expansion of x(1+2x)/(1+x)^2.