This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A039809 #27 Jul 23 2021 02:42:46
%S A039809 1,1,2,5,12,32,83,223,594,1600,4297,11589,31216,84212,227091,612712,
%T A039809 1652913,4459962,12033405,32469682,87611105,236402465,637884103,
%U A039809 1721218224,4644392797,12532091909,33815653370,91245738923
%N A039809 For n > 1, a(n) doubles under the transform T, where Ta is the matrix product of partition triangle A008284 with a, with a(1) = 1.
%F A039809 a(1) = 1 and a(n) = Sum_{i=1..n-1} A008284(n, i)*a(i) for n >= 2 (because 2*a(n) = Sum_{i=1..n} A008284(n,i)*a(i) for n >= 2).
%F A039809 a(n+1) = Sum_{k=0..n} A081719(n,k). - _Philippe Deléham_, Sep 30 2006
%F A039809 G.f.: (1/2) * ( x + Sum_{n>=1} a(n) * x^n / Product_{j=1..n} (1 - x^j) ). - _Ilya Gutkovskiy_, Jul 22 2021
%e A039809 So a(7) = T(7,1)*a(1) + T(7,2)*a(2) + ... + T(7,6)*a(6) = 1*1 + 3*1 + 4*2 + 3*5 + 2*12 + 1*32 = 1 + 3 + 8 + 15 + 24 + 32 = 83, where T(n,k) = A008284(n,k).
%o A039809 (PARI) P(n, k) = #partitions(n-k, k); /* A008284 */
%o A039809 lista(nn) = {my(a=vector(nn)); a[1]=1; for(n=2, nn, a[n] = sum(i=1, n-1, P(n,i)*a[i])); a;} \\ _Petros Hadjicostas_, May 30 2020
%Y A039809 Cf. A008284, A039808, A081719.
%K A039809 nonn,eigen
%O A039809 1,3
%A A039809 _Christian G. Bower_, Feb 15 1999
%E A039809 Various sections edited by _Petros Hadjicostas_, May 30 2020