A378816 -id:A378816 - cp's OEIS frontend

A378817 Hankel sequence transform of A378816.

-1, -5, 6, -10, 11, -1, 1, 17, -18, 22, -23, 1, -1, -29, 30, -34, 35, -1, 1, 41, -42, 46, -47, 1, -1, -53, 54, -58, 59, -1, 1, 65, -66, 70, -71, 1, -1, -77, 78, -82, 83, -1, 1, 89, -90, 94, -95, 1, -1, -101, 102, -106, 107, -1, 1, 113, -114, 118, -119

Offset: 0

Thomas Scheuerle, Dec 08 2024

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-3,0,2,0,-1).

Cf. A378783, A378816.

LinearRecurrence[{0, 2, 0, -3, 0, 2, 0, -1}, {-1, -5, 6, -10, 11, -1, 1, 17}, 100] (* Paolo Xausa, Jan 28 2025 *)

a(n) =  (1+(2*n+3-n%3)*((n%6)*((n+1)%6)>0))*(-1)^(n+1+ceil(n/6))

G.f.: (-1 - 5*x + 8*x^2 - 4*x^4 + 4*x^5 - x^6 - x^7)/(1 - x^2 + x^4)^2.
a(n) = 2*a(n-2) - 3*a(n-4) + 2*a(n-6) - a(n-8).
a(12*n) = -1.
a(5+12*n) = -1.
a(6+12*n) = 1.
a(11+12*n) = 1.
a(1+12*n) = -5-24*n.
a(2+12*n) = 6+24*n.
a(3+12*n) = -10-24*n.
a(4+12*n) = 11+24*n.
a(7+12*n) = 17+24*n.
a(8+12*n) = -18-24*n.
a(9+12*n) = 22+24*n.
a(10+12*n) = -23-24*n.

A378783 Triangular array T(n,k) read by rows: T(n, k) = c_k(n+1). The sequence c_k(m) has the ordinary generating function C_k(x) which satisfies C_k(x) = 1/(1+C_k(x)*Sum_{t=0..k} x^(t+1)).

Original entry on oeis.org

-1, 2, 1, -5, -1, -2, 14, 1, 5, 4, -42, -1, -12, -8, -9, 132, 1, 29, 18, 22, 21, -429, -1, -73, -43, -54, -50, -51, 1430, 1, 190, 105, 135, 124, 128, 127, -4862, -1, -505, -262, -345, -315, -326, -322, -323, 16796, 1, 1363, 666, 896, 813, 843, 832, 836, 835

Offset: 0

Thomas Scheuerle, Dec 07 2024

			Triangle begins:
  [0]    -1
  [1]     2,  1
  [2]    -5, -1,   -2
  [3]    14,  1,    5,    4
  [4]   -42, -1,  -12,   -8,   -9
  [5]   132,  1,   29,   18,   22,   21
  [6]  -429, -1,  -73,  -43,  -54,  -50,  -51
  [7]  1430,  1,  190,  105,  135,  124,  128,  127
  [8] -4862, -1, -505, -262, -345, -315, -326, -322, -323
.

Cf. A001006, A000108, A152171, A166587, A168491, A378816.

T[n_,k_]:=SeriesCoefficient[(2 / (Sqrt[1+4*Sum[x^(t+1),{t,0,k}]] + 1) - 1)/x,{x,0,n}];Table[T[n,k],{n,0,9},{k,0,n}]//Flatten (* Stefano Spezia, Dec 08 2024 *)

column(n, max_n) = { my(s = 1,x = 'x,cu); for(k = 0, max_n-1, cu = cu+polcoeff(1/s+O(x^(k+1)), k, x); cu = cu-polcoeff(1/s+O(x^(k+1)), k-1-n, x); s = s+cu*x^(k+1)); Vec(1/s+O(x^max_n)) };
T(n, k) = column(k, n+2)[n+2]
T(n, k) = polcoeff(2 / (sqrt(1+4*x*sum(t=0, k, x^t)) + 1) + O(x^(n+2)), n+1, x)

G.f. column k: (2 / (sqrt(1+4*Sum_{t=0..k}x^(t+1)) + 1) - 1)/x.
T(n, 0) = (-1)^(n+1)*Catalan(n+1) = A168491(n+1).
T(n, 2) = (-1)^(n+1)*A152171(n+1).
T(n, n) = (-1)^(n+1)*A001006(n) = -A166587(n+1).
A378816(n) = Limit_{k->oo} (T(k, k-n) - T(k, k-n-1)).

cp's OEIS Frontend

A378817 Hankel sequence transform of A378816.

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