A073377 Seventh convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

1, 8, 52, 264, 1194, 4872, 18516, 66264, 226083, 740608, 2344232, 7202416, 21562164, 63090288, 180884088, 509245776, 1410356133, 3848340312, 10359516684, 27544099704, 72406891326, 188356187448

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (8,-12,-56,154,168,-700,-328,1791,656,-2800,-1344, 2464,1792,-768,-1024,-256).

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073376(k).

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

a(n) = ((328247920 +332102604*n +131833680*n^2 +26450901*n^3 +2844099*n^4 + 156087*n^5 +3429*n^6)*(n+1)*U(n+1) + 2(141143240 +150941694*n +62335731*n^2 + 12873492*n^3 +1414314*n^4 +78894*n^5 +1755*n^6)*(n+2)*U(n))/(7!*3^11) with U(n) = A001045(n+1), n>=0.

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

E.g.f.: (1/(7!*3^12))*( 4096*(596225 +4177950*x +7304850*x^2 +5109300*x^3 +1691550*x^4 +278964*x^5 +21924*x^6 +648*x^7)*exp(2*x) + (236325040 -333132240*x +158026680*x^2 -34637400*x^3 +3921750*x^4 -234738*x^5 +6993*x^6 -81*x^7)*exp(-x) ). - G. C. Greubel, Sep 29 2022

Mathematica

Table[(2^(n+8)*(9539600 +17240268*n +11555460*n^2 +3849489*n^3 +703080*n^4 +71442*n^5 +3780*n^6 +81*n^7) +(-1)^n*(236325040 +225702732*n +87290028*n^2 +17880849*n^3 +2109240*n^4 +144018*n^5 +5292*n^6 +81*n^7))/(7!*3^12), {n, 0, 60}] (* G. C. Greubel, Sep 29 2022 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1-2*x))^8 )); // G. C. Greubel, Sep 29 2022
(SageMath)
def A073377(n): return (2^(n+8)*(9539600 +17240268*n +11555460*n^2 +3849489*n^3 +703080*n^4 +71442*n^5 +3780*n^6 +81*n^7) +(-1)^n*(236325040 +225702732*n +87290028*n^2 +17880849*n^3 +2109240*n^4 +144018*n^5 +5292*n^6 +81*n^7))/(factorial(7)*3^12)
[A073377(n) for n in range(40)] # G. C. Greubel, Sep 29 2022

Crossrefs

Eighth (m=7) column of triangle A073370.

Cf. A001045, A073376.

Sequence in context: A107584 A323940 A027225 * A055283 A193427 A022732

Adjacent sequences: A073374 A073375 A073376 * A073378 A073379 A073380

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 02 2002

Status

approved