A117413 -id:A117413 - cp's OEIS frontend

A117411 Skew triangle associated to the Euler numbers.

1, 0, 1, 0, -4, 1, 0, 0, -12, 1, 0, 0, 16, -24, 1, 0, 0, 0, 80, -40, 1, 0, 0, 0, -64, 240, -60, 1, 0, 0, 0, 0, -448, 560, -84, 1, 0, 0, 0, 0, 256, -1792, 1120, -112, 1, 0, 0, 0, 0, 0, 2304, -5376, 2016, -144, 1, 0, 0, 0, 0, 0, -1024, 11520, -13440, 3360, -180, 1, 0, 0, 0, 0, 0, 0, -11264, 42240, -29568, 5280, -220, 1

Offset: 0

Paul Barry, Mar 13 2006

Inverse is A117414. Row sums of the inverse are the Euler numbers A000364.

Triangle, read by rows, given by [0,-4,4,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2009

			Triangle begins
  1;
  0,  1;
  0, -4,   1;
  0,  0, -12,   1;
  0,  0,  16, -24,    1;
  0,  0,   0,  80,  -40,     1;
  0,  0,   0, -64,  240,   -60,      1;
  0,  0,   0,   0, -448,   560,    -84,      1;
  0,  0,   0,   0,  256, -1792,   1120,   -112,      1;
  0,  0,   0,   0,    0,  2304,  -5376,   2016,   -144,      1;
  0,  0,   0,   0,    0, -1024,  11520, -13440,   3360,   -180,    1;
  0,  0,   0,   0,    0,     0, -11264,  42240, -29568,   5280, -220,    1;
  0,  0,   0,   0,    0,     0,   4096, -67584, 126720, -59136, 7920, -264, 1;

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

Cf. A000364, A006495 (row sums), A098158, A117413, A117414.

Cf. A000217, A000332, A000579, A000581, A262710.

A117411:= func< n,k | (-4)^(n-k)*(&+[Binomial(n,k-j)*Binomial(j,n-k): j in [0..n-k]]) >;
[A117411(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Sep 07 2022

T[n_,k_]:= T[n,k]= (-4)^(n-k)*Sum[Binomial[n, k-j]*Binomial[j, n-k], {j,0,n-k}];
Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 07 2022 *)

def A117411(n,k): return (-4)^(n-k)*sum(binomial(n,k-j)*binomial(j,n-k) for j in (0..n-k))
flatten([[A117411(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Sep 07 2022

Sum_{k=0..n} T(n, k) = A006495(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A117413(n).
T(n, k) = (-4)^(n-k)*Sum_{j=0..n-k} C(n,k-j)*C(j,n-k).
G.f.: (1-x*y)/(1-2x*y+x^2*y(y+4)). - Paul Barry, Mar 14 2006
T(n, k) = (-4)^(n-k)*A098158(n,k). - Philippe Deléham, Nov 01 2009
T(n, k) = 2*T(n-1,k-1) - 4*T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 31 2013
From G. C. Greubel, Sep 07 2022: (Start)
T(n, n) = 1.
T(n, n-1) = -4*A000217(n-1), n >= 1.
T(n, n-2) = (-4)^2 * A000332(n), n >= 2.
T(n, n-3) = (-4)^3 * A000579(n), n >= 3.
T(n, n-4) = (-4)^4 * A000581(n), n >= 4.
T(2*n, n) = A262710(n). (End)

A052922 Expansion of 1/(1 - 2*x^3 - x^4).

Original entry on oeis.org

1, 0, 0, 2, 1, 0, 4, 4, 1, 8, 12, 6, 17, 32, 24, 40, 81, 80, 104, 202, 241, 288, 508, 684, 817, 1304, 1876, 2318, 3425, 5056, 6512, 9168, 13537, 18080, 24848, 36242, 49697, 67776, 97332, 135636, 185249, 262440, 368604, 506134, 710129, 999648, 1380872

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 907
Index entries for linear recurrences with constant coefficients, signature (0,0,2,1).

Cf. A117413 (bijection).

a:=[1,0,0,2];; for n in [5..50] do a[n]:=2*a[n-3]+a[n-4]; od; a; # G. C. Greubel, Oct 16 2019

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1 -2*x^3 -x^4) )); // G. C. Greubel, Oct 16 2019

spec := [S,{S=Sequence(Prod(Z,Z,Union(Z,Z,Prod(Z,Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..50);
seq(coeff(series(1/(1 -2*x^3 -x^4), x, n+1), x, n), n = 0..50); # G. C. Greubel, Oct 16 2019
# alternative
A052922 := proc(n)
    if n <=3 then
        op(n+1,[1,0,0,2]) ;
    else
        2*procname(n-3)+procname(n-4) ;
    end if;
end proc:
seq(A052922(n),n=0..30) ; # R. J. Mathar, Nov 22 2024

LinearRecurrence[{0,0,2,1}, {1,0,0,2}, 50] (* G. C. Greubel, Oct 16 2019 *)
CoefficientList[Series[1/(1-2x^3-x^4),{x,0,50}],x] (* Harvey P. Dale, Nov 04 2024 *)

my(x='x+O('x^50)); Vec(1/(1 -2*x^3 -x^4)) \\ G. C. Greubel, Oct 16 2019

def A052922_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/(1 -2*x^3 -x^4)).list()
A052922_list(50) # G. C. Greubel, Oct 16 2019

G.f.: 1/(1 - 2*x^3 - x^4).
a(n) = 2*a(n-3) + a(n-4), with a(0)=1, a(1)=0, a(2)=0, a(3)=2.
a(n) = Sum_{alpha=RootOf(-1+2*z^3+z^4)} (1/86)*(4 +26*alpha -3*alpha^2 -6*alpha^3)*alpha^(-1-n).

More terms from James Sellers, Jun 05 2000

cp's OEIS Frontend

A117411 Skew triangle associated to the Euler numbers.

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