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A045622 Convolution of A000108 (Catalan numbers) with A045543. 2

%I

%S 1,25,362,3973,36646,299530,2238676,15613741,103054094,650194974,

%T 3950996556,23257207714,133217073276,745218012084,4083224828328,

%U 21966983072637,116268166691358,606474982072982,3122157367765788

%N Convolution of A000108 (Catalan numbers) with A045543.

%C Also convolution of A045530 with A000984 (central binomial coefficients); also convolution of A045505 with A000302 (powers of 4).

%H G. C. Greubel, <a href="/A045622/b045622.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = binomial(n+6, 5)*(4^(n+1) - A000984(n+6)/A000984(5))/2, A000984(n) = binomial(2*n, n).

%F G.f.: x*c(x)/(1-4*x)^6, where c(x) = g.f. for Catalan numbers.

%p seq(coeff(series((1-sqrt(1-4*x))/(2*(1-4*x)^6), x, n+1), x, n), n = 0..40); # _G. C. Greubel_, Jan 13 2020

%t CoefficientList[Series[(1-Sqrt[1-4*x])/(2*x*(1-4*x)^6), {n,0,40}], x] (* _G. C. Greubel_, Jan 13 2020 *)

%o (PARI) my(x='x+O('x^40)); Vec((1-sqrt(1-4*x))/(2*(1-4*x)^6)) \\ _G. C. Greubel_, Jan 13 2020

%o (MAGMA) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-4*x))/(2*(1-4*x)^6) )); // _G. C. Greubel_, Jan 13 2020

%o (Sage)

%o def A045622_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (1-sqrt(1-4*x))/(2*(1-4*x)^6) ).list()

%o A045622_list(40) # _G. C. Greubel_, Jan 13 2020

%K easy,nonn

%O 1,2

%A _Wolfdieter Lang_

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Last modified August 7 17:44 EDT 2020. Contains 336278 sequences. (Running on oeis4.)