A213837 Principal diagonal of the convolution array A213836.

1, 13, 52, 134, 275, 491, 798, 1212, 1749, 2425, 3256, 4258, 5447, 6839, 8450, 10296, 12393, 14757, 17404, 20350, 23611, 27203, 31142, 35444, 40125, 45201, 50688, 56602, 62959, 69775, 77066, 84848, 93137

Offset: 1

Author

Clark Kimberling, Jul 04 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 (4,-6,4,-1).

Crossrefs

Cf. A000384, A002412, A213836, A220084 (for a list of numbers of the form n*P(k,n)-(n-1)*P(k,n-1), where P(k,n) is the n-th k-gonal pyramidal number).

Programs

Mathematica

(See A213836.)
LinearRecurrence[{4,-6,4,-1},{1,13,52,134},40] (* Harvey P. Dale, Aug 07 2025 *)

Formula

a(n) = n*(5 - 15*n + 16*n^2)/6.

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

G.f.: f(x)/g(x), where f(x) = x*(1 + 9*x + 6*x^2) and g(x) = (1-x)^4.

a(n) = n*A002412(n) - (n-1)*A002412(n-1). [Bruno Berselli, Dec 11 2012]

a(n) = n*A000384(n) + sum( A000384(i), i=0..n-1 ). [Bruno Berselli, Dec 18 2013]