cp's OEIS Frontend

A307972 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5*A(x)^2.

%I A307972 #19 Nov 27 2022 15:29:09
%S A307972 1,1,1,1,1,1,2,3,4,5,6,9,14,21,30,41,58,86,130,195,286,416,612,915,
%T A307972 1380,2076,3102,4627,6932,10452,15818,23931,36148,54600,82642,125435,
%U A307972 190724,290116,441282,671512,1023052,1560780,2383578,3642117,5567202,8514254,13031192,19960712
%N A307972 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5*A(x)^2.
%C A307972 Shifts 5 places left when convolved with itself.
%H A307972 Seiichi Manyama, <a href="/A307972/b307972.txt">Table of n, a(n) for n = 0..5000</a>
%F A307972 G.f.: 1/(1 - x/(1 - x^5/(1 - x^5/(1 - x/(1 - x^5/(1 - x^5/(1 - x/(1 - x^5/(1 - x^5/(1 - ...)))))))))), a continued fraction.
%F A307972 Recurrence: a(n+5) = Sum_{k=0..n} a(k)*a(n-k).
%e A307972 G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + 6*x^10 + ...
%p A307972 a:= proc(n) option remember; `if`(n<5, 1,
%p A307972       add(a(j)*a(n-5-j), j=0..n-5))
%p A307972     end:
%p A307972 seq(a(n), n=0..50);  # _Alois P. Heinz_, May 08 2019
%t A307972 terms = 47; A[_] = 0; Do[A[x_] = 1 + x + x^2 + x^3 + x^4 + x^5 A[x]^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
%t A307972 a[n_] := a[n] = Sum[a[k] a[n - k - 5], {k, 0, n - 5}]; a[0] = a[1] = a[2] = a[3] = a[4] = 1; Table[a[n], {n, 0, 47}]
%o A307972 (SageMath)
%o A307972 @CachedFunction
%o A307972 def a(n): # a = A307972
%o A307972     if (n<5): return 1
%o A307972     else: return sum(a(k)*a(n-k-5) for k in range(n-4))
%o A307972 [a(n) for n in range(51)] # _G. C. Greubel_, Nov 26 2022
%Y A307972 Cf. A000108, A007477, A307970, A307971.
%K A307972 nonn
%O A307972 0,7
%A A307972 _Ilya Gutkovskiy_, May 08 2019