A350498 Convolution of triangular numbers with every third number of Narayana's Cows sequence.

0, 1, 7, 31, 114, 385, 1250, 3987, 12619, 39810, 125425, 394955, 1243433, 3914383, 12322293, 38789576, 122105944, 384377494, 1209981891, 3808901216, 11990036895, 37743426054, 118812495000, 374009739009, 1177344897390, 3706162867858, 11666626518622, 36725362368682, 115607732787126, 363921470561515

Offset

1,3

Comments

This is the convolution of N(3*n-1) with t(n); in other words, a(n) = Sum_{i=1..n} N(3*i-1)*t(n-i) where N(k)=A000930(k) is the k-th number in Narayana's Cows sequence and t(k)=A000217(k) is the k-th triangular number.

References

G. Dresden and M. Tulskikh, "Convolutions of Sequences with Single-Term Signature Differences", preprint.

Links

Table of n, a(n) for n=1..30.

Index entries for linear recurrences with constant coefficients, signature (7,-18,23,-16,6,-1).

Formula

a(n) = N(3*n-1) - A000217(n) where N(k)=A000930(k).

G.f.: x^2/((1 - x)^3 * (1 - 4*x + 3*x^2 - x^3)).

a(n) = A052529(n)-A000217(n), n>0. - R. J. Mathar, Aug 17 2022

Example

For n=4, a(4) = N(2)*t(3) + N(5)*t(2) + N(8)*t(1) + N(11)*t(0) = 1*6 + 4*3 + 13*1 + 41*0 = 31, where N(k)=A000930(k) and t(k)=A000217(k).

Mathematica

CoefficientList[
Series[x/((-1 + x)^3 (-1 + 4 x - 3 x^2 + x^3)), {x, 0, 30}], x]

Crossrefs

Cf. A000217, A000930.

Sequence in context: A055580 A364635 A097786 * A197649 A006458 A091344

Adjacent sequences: A350495 A350496 A350497 * A350499 A350500 A350501

Keyword

nonn,easy

Author

Greg Dresden, Jan 04 2022

Status

approved