A213581 Antidiagonal sums of the convolution array A213571.

1, 8, 36, 124, 367, 988, 2498, 6048, 14197, 32576, 73472, 163508, 360027, 785908, 1703294, 3669240, 7863393, 16776120, 35650300, 75495980, 159381831, 335542348, 704640826, 1476392464, 3087004877, 6442447728, 13421769208

Offset

1,2

Links

Clark Kimberling, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (8,-26,44,-41,20,-4).

Formula

a(n) = 8*a(n-1) - 26*a(n-2) + 44*a(n-3) - 41*a(n-4) + 20*a(n-5) - 4*a(n-6).

G.f.: f(x)/g(x), where f(x) = x*(1 - 2*x^2) and g(x) = (1 - x)^4*(1 - 2*x)^2.

a(n) = 8 +(n-2)*2^(n+2) -(n-2)*n*(n+5)/6. - Bruno Berselli, Jul 09 2012

Mathematica

(* First Program *)
b[n_]:= n; c[n_]:= -1 + 2^n;
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213571 *)
d = Table[t[n, n], {n, 1, 40}] (* A213572 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213581 *)
(* Second program *)
Table[2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6, {n, 35}] (* G. C. Greubel, Jul 26 2019 *)

Prog

(PARI) vector(35, n, 2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6) \\ G. C. Greubel, Jul 26 2019
(Magma) [2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6: n in [1..35]]; // G. C. Greubel, Jul 26 2019
(Sage) [2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6 for n in (1..35)] # G. C. Greubel, Jul 26 2019
(GAP) List([1..35], n-> 2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6); # G. C. Greubel, Jul 26 2019

Crossrefs

Cf. A213571, A213500.

Sequence in context: A054470 A347751 A341222 * A276279 A210379 A131123

Adjacent sequences: A213578 A213579 A213580 * A213582 A213583 A213584

Keyword

nonn,easy

Author

Clark Kimberling, Jun 19 2012

Status

approved