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A073376
Sixth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
3
1, 7, 42, 196, 826, 3150, 11256, 38004, 122787, 381997, 1151458, 3376968, 9671284, 27123292, 74669472, 202181112, 539342181, 1419492627, 3690464106, 9487902396, 24143758254, 60861096714
OFFSET
0,2
LINKS
Index entries for linear recurrences with constant coefficients, signature (7,-7,-49,91,161,-357,-363,714,644,-728,-784,224, 448,128).
FORMULA
a(n) = Sum_{k=0..n} b(k) * c(n-k), with b(k) = A001045(k+1) and c(k) = A073375(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+6, 6) * binomial(n-k, k) * 2^k.
a(n) = ((1+n)*(4884880 +4449396*n +1525272*n^2 +247653*n^3 +19152*n^4 +567*n^5)* U(n+1) + 2*(2+n)*(2321720 +2182242*n +765993*n^2 +126621*n^3 +9927*n^4 +297*n^5 )*U(n))/(6!*3^9) with U(n) = A001045(n+1), n>=0.
G.f.: 1/(1-(1+2*x)*x)^7 = 1/((1+x)*(1-2*x))^7.
E.g.f.: (1/(6!*3^10))*( 4096*(9305 +56535*x +83745*x^2 +47700*x^3 +12060*x^4 +1350*x^5 +54*x^6)*exp(2*x) + (4402000 -5784960*x +2454120*x^2 -457920*x^3 +41130*x^4 -1728*x^5 +27*x^6)*exp(-x)). - G. C. Greubel, Sep 29 2022
MATHEMATICA
Table[(2^(n+7)*(297760 +500640*n +302778*n^2 +87255*n^3 +12915*n^4 +945*n^5 +27*n^6) +(-1)^n*(4402000 +4038132*n +1453788*n^2 +265545*n^3 +26145*n^4 +1323*n^5 +27*n^6))/(6!*3^10), {n, 0, 40}] (* G. C. Greubel, Sep 29 2022 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1-2*x))^7 )); // G. C. Greubel, Sep 29 2022
(SageMath)
def A073376(n): return (2^(n+7)*(297760 +500640*n +302778*n^2 +87255*n^3 +12915*n^4 +945*n^5 +27*n^6) +(-1)^n*(4402000 +4038132*n +1453788*n^2 +265545*n^3 +26145*n^4 +1323*n^5 +27*n^6))/(factorial(6)*3^10)
[A073376(n) for n in range(40)] # G. C. Greubel, Sep 29 2022
CROSSREFS
Seventh (m=6) column of triangle A073370.
Sequence in context: A008526 A057425 A248329 * A094429 A246434 A255614
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 02 2002
STATUS
approved