A213583 Principal diagonal of the convolution array A213582.

1, 9, 38, 120, 327, 819, 1948, 4482, 10085, 22341, 48930, 106236, 229075, 491175, 1048184, 2227782, 4718097, 9960921, 20970910, 44039520, 92273951, 192937179, 402652308, 838859850, 1744829437, 3623877549, 7516191578, 15569255172, 32212253355, 66571991631

Offset

1,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (7,-19,25,-16,4).

Formula

a(n) = 7*a(n-1) - 19*a(n-2) + 25*a(n-3) - 16*a(n-4) + 4*a(n-5).

G.f.: x*(1 + 2*x - 6*x^2) / ((1 - x)^3*(1 - 2*x)^2).

a(n) = (n+1)*(2^(n+2) - 3*n -4)/2. - Colin Barker, Nov 04 2017

E.g.f.: (4*(1+2*x)*exp(2*x) - (3*x^2+10*x+4)*exp(x))/2. - G. C. Greubel, Jul 08 2019

Mathematica

(* First program *)
b[n_]:= 2^n - 1; c[n_]:= 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}]] (* A213582 *)
r[n_]:= Table[T[n, k], {k, 40}] (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213583 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A156928 *)
(* Second program *)
LinearRecurrence[{7, -19, 25, -16, 4}, {1, 9, 38, 120, 327}, 40] (* Harvey P. Dale, Apr 06 2013 *)
Table[(n+1)*(2^(n+2)-3*n-4)/2, {n, 40}] (* G. C. Greubel, Jul 08 2019 *)

Prog

(PARI) Vec(x*(1 + 2*x - 6*x^2) / ((1 - x)^3*(1 - 2*x)^2) + O(x^40)) \\ Colin Barker, Nov 04 2017
(PARI) vector(40, n, (n+1)*(2^(n+2) -3*n-4)/2) \\ G. C. Greubel, Jul 08 2019
(Magma) [(n+1)*(2^(n+2) -3*n-4)/2: n in [1..40]]; // G. C. Greubel, Jul 08 2019
(Sage) [(n+1)*(2^(n+2) -3*n-4)/2 for n in (1..40)] # G. C. Greubel, Jul 08 2019
(GAP) List([1..40], n-> (n+1)*(2^(n+2) -3*n-4)/2) # G. C. Greubel, Jul 08 2019

Crossrefs

Cf. A213500, A213582.

Sequence in context: A120780 A071229 A071238 * A343521 A050854 A053181

Adjacent sequences: A213580 A213581 A213582 * A213584 A213585 A213586

Keyword

nonn,easy

Author

Clark Kimberling, Jun 19 2012

Status

approved