A366017 - OEIS

%I #6 Sep 26 2023 17:01:57

%S 0,1,9,132,2365,47169,1005564,22431720,517122117,12222124035,

%T 294569159313,7212098118888,178877944712844,4484938858752940,

%U 113488477622130600,2894560146756466320,74335973069605120725,1920587845828953301479,49886703842977713177723,1301959618949870922531300,34123873581608909988904245

%N G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 5 * A(x)).

%C Reversion of g.f. for octagonal pyramidal numbers (with signs).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SeriesReversion.html">Series Reversion</a>

%F a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(4*n,n-k-1) * 5^k for n > 0.

%t nmax = 20; A[_] = 0; Do[A[x_] = x (1 + A[x])^4/(1 - 5 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t CoefficientList[InverseSeries[Series[x (1 - 5 x)/(1 + x)^4, {x, 0, 20}], x], x]	

%t Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[4 n, n - k - 1] 5^k, {k, 0, n - 1}], {n, 1, 20}]]

%Y Cf. A002293, A002414, A064089, A365754, A365818, A366014, A366015, A366016.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Sep 26 2023