A213554 Principal diagonal of the convolution array A213553.

1, 43, 334, 1406, 4271, 10577, 22764, 44220, 79437, 134167, 215578, 332410, 495131, 716093, 1009688, 1392504, 1883481, 2504067, 3278374, 4233334, 5398855, 6807977, 8497028, 10505780, 12877605, 15659631, 18902898, 22662514

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n) = n*(39*n^4 + 15*n^3 - 25*n^2 + 1)/30.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

G.f.: x*(1 + 37*x + 91*x^2 + 27*x^3)/(1 - x)^6.

Mathematica

(* First program *)
b[n_]:= n; c[n_]:= n^3;
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}]  (* A213553 *)
d = Table[T[n, n], {n, 1, 40}] (* A213554 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A101089 *)
(* Second program *)
Table[(39n^5+15n^4-25n^3+n)/30, {n, 30}] (* or *) LinearRecurrence[ {6, -15, 20, -15, 6, -1}, {1, 43, 334, 1406, 4271, 10577}, 30] (* Harvey P. Dale, Jan 15 2013 *)

Prog

(PARI) vector(30, n, n*(39*n^4 +15*n^3 -25*n^2 +1)/30) \\ G. C. Greubel, Jul 31 2019
(Magma) [n*(39*n^4 +15*n^3 -25*n^2 +1)/30: n in [1..30]]; // G. C. Greubel, Jul 31 2019
(Sage) [n*(39*n^4 +15*n^3 -25*n^2 +1)/30 for n in (1..30)] # G. C. Greubel, Jul 31 2019
(GAP) List([1..30], n-> n*(39*n^4 +15*n^3 -25*n^2 +1)/30); # G. C. Greubel, Jul 31 2019

Crossrefs

Cf. A213500, A213553.

Sequence in context: A364320 A346575 A217621 * A164783 A291861 A142502

Adjacent sequences: A213551 A213552 A213553 * A213555 A213556 A213557

Keyword

nonn,easy

Author

Clark Kimberling, Jun 17 2012

Status

approved