A073394 Seventh convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

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

Author

Wolfdieter Lang, Aug 02 2002

Keywords

• nonn
• easy

Examples

			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
		

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).

Crossrefs

Eighth (m=7) column of triangle A073387.

Cf. A002605, A073393.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^8 )); // G. C. Greubel, Oct 06 2022

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 *)

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

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k) with b(k) = A002605(k) 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+1) + 4*(235900 + 375554*n + 207009*n^2 + 54174*n^3 + 7318*n^4 + 492*n^5 + 13*n^6)*(n+2)*U(n))/(2^8*3^6*5*7), with U(n) = A002605(n), n >= 0.

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

Extensions

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