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.
1, 1, 2, 5, 12, 32, 83, 223, 594, 1600, 4297, 11589, 31216, 84212, 227091, 612712, 1652913, 4459962, 12033405, 32469682, 87611105, 236402465, 637884103, 1721218224, 4644392797, 12532091909, 33815653370, 91245738923
Offset: 1
Examples
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).
Programs
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PARI
P(n, k) = #partitions(n-k, k); /* A008284 */ 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
Formula
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).
a(n+1) = Sum_{k=0..n} A081719(n,k). - Philippe Deléham, Sep 30 2006
G.f.: (1/2) * ( x + Sum_{n>=1} a(n) * x^n / Product_{j=1..n} (1 - x^j) ). - Ilya Gutkovskiy, Jul 22 2021
Extensions
Various sections edited by Petros Hadjicostas, May 30 2020