A213572 Principal diagonal of the convolution array A213571.

1, 13, 82, 406, 1809, 7659, 31588, 128476, 518611, 2084809, 8361918, 33497010, 134094757, 536608663, 2146926472, 8588754808, 34357247847, 137433710421, 549744803650, 2199000186670, 8796044787481, 35184271425283

Offset

1,2

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (11,-47,101,-116,68,-16).

Formula

a(n) = (2^(n+2)*(2^n-1) - (2^(n+1) + n + 1)*n)/2.

a(n) = 11*a(n-1) - 47*a(n-2) + 101*a(n-3) - 116*a(n-4) + 68*a(n-5) - 16*a(n-6).

G.f.: f(x)/g(x), where f(x) = x*(1 + 2*x - 14*x^2 + 14*x^3) and g(x) = (1 - 4*x)*((1 - x)^3)*(1 - 2*x)^2.

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 *)
(* Additional programs *)
Table[2^(2*n+1) -2^n*(n+2)-Binomial[n+1, 2], {n, 30}] (* G. C. Greubel, Jul 25 2019 *)

Prog

(PARI) vector(30, n, 2^(2*n+1) -2^n*(n+2) -binomial(n+1, 2)) \\ G. C. Greubel, Jul 25 2019
(Magma) [2^(2*n+1) -2^n*(n+2) -Binomial(n+1, 2): n in [1..30]]; // G. C. Greubel, Jul 25 2019
(Sage) [2^(2*n+1) -2^n*(n+2) -binomial(n+1, 2) for n in (1..30)] # G. C. Greubel, Jul 25 2019
(GAP) List([1..30], n-> 2^(2*n+1) -2^n*(n+2) -Binomial(n+1, 2)); # G. C. Greubel, Jul 25 2019

Crossrefs

Cf. A213571, A213500.

Sequence in context: A082203 A367118 A101102 * A142085 A163688 A274398

Adjacent sequences: A213569 A213570 A213571 * A213573 A213574 A213575

Keyword

nonn,easy

Author

Clark Kimberling, Jun 19 2012

Status

approved