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

Author

Clark Kimberling, Jun 19 2012

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A213500, A213582.

Programs

GAP

List([1..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

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 *)

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

Sage

[(n+1)*(2^(n+2) -3*n-4)/2 for n in (1..40)] # G. C. Greubel, Jul 08 2019

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