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

Author

Clark Kimberling, Jun 17 2012

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A213500, A213553.

Programs

GAP

List([1..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

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 *)

PARI

vector(30, n, n*(39*n^4 +15*n^3 -25*n^2 +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

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.