A073394 Eighth column of the convolution triangle A073387(n+7, 7), for n >= 0.

1, 16, 160, 1248, 8304, 49344, 269184, 1372800, 6628512, 30584576, 135804416, 583471616, 2436145920, 9919484928, 39503038464, 154230921216, 591550292736, 2232748892160, 8305370185728, 30486351396864, 110551407403008, 396424924397568, 1406924861276160, 4945692873129984, 17231635316293632

Offset

0,2

Comments

Original name: Seventh convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself. (See the comment in A073387).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (16,-96,224,112,-1344,896,3712,-3168,-7424,3584,10752,1792,-7168,-6144,-2048,-256).

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k) with b(k) = A002605(k+1) and c(k) = A073393(k).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+7, 7)*binomial(n-k, k)*2^(n-k).

a(n) = ((2322320 + 2869040*n + 1379232*n^2 + 332247*n^3 + 42533*n^4 + 2757*n^5 + 71*n^6)*(n+1)*U(n+2) + 4*(235900 + 375554*n + 207009*n^2 + 54174*n^3 + 7318*n^4 + 492*n^5 + 13*n^6)*(n+2)*U(n+1))/(2^8*3^6*5*7), with U(n) = A002605(n), n >= 0.

G.f.: 1/(1-2*x*(1+x))^8.

Example

G.f. = 1 + 16*x + 160*x^2 + 1248*x^3 + ... + 154230921216*x^15 + 591550292736*x^16 + 2232748892160*x^17 + 8305370185728*x^18 + ... - Zerinvary Lajos, Jun 03 2009

Mathematica

CoefficientList[Series[1/(1-2*x-2*x^2)^8, {x, 0, 30}], x] (* G. C. Greubel, Oct 06 2022 *)
LinearRecurrence[{16, -96, 224, 112, -1344, 896, 3712, -3168, -7424, 3584, 10752, 1792, -7168, -6144, -2048, -256}, {1, 16, 160, 1248, 8304, 49344, 269184, 1372800, 6628512, 30584576, 135804416, 583471616, 2436145920, 9919484928, 39503038464, 154230921216}, 30] (* Harvey P. Dale, Nov 21 2023 *)

Prog

(SageMath) taylor( 1/(1-2*x-2*x^2)^8, x, 0, 26).list() # Zerinvary Lajos, Jun 03 2009; modified by G. C. Greubel, Oct 06 2022
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^8 )); // G. C. Greubel, Oct 06 2022

Crossrefs

Cf. A002605, A073387, A073393.

Sequence in context: A144453 A121036 A224058 * A038846 A079767 A079768

Adjacent sequences: A073391 A073392 A073393 * A073395 A073396 A073397

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 02 2002

Extensions

Terms a(19) onward added by G. C. Greubel, Oct 06 2022

New name from Wolfdieter Lang, May 06 2026

Status

approved