|
%I #22 Mar 18 2022 10:27:47
%S 1,2,3,4,7,8,9,14,20,20,21,32,43,46,51,62,71,82,107,136,145,136,144,
%T 200,280,316,296,294,359,456,535,576,591,650,820,1020,1078,990,963,
%U 1160,1541,1950,2225,2244,2034,1892,2211,3024,3866,4260,4207,4066,4150,4630,5617
%N Convolution of A007318 (Pascal's sequence) with itself .
%H Paul D. Hanna, <a href="/A152037/b152037.txt">Table of n, a(n) for n = 0..1035</a>
%F G.f.: [Sum_{n>=0} x^(n*(n+1)/2) * (1+x)^n ]^2. [_Paul D. Hanna_, Apr 18 2012]
%F G.f.: [Sum_{n>=0} (1+x)^n*x^n * Product_{k=1..n} (1 - (1+x)*x^(2*k-1)) / (1 - (1+x)*x^(2*k)) ]^2. [_Paul D. Hanna_, Apr 18 2012]
%p g:= proc(n) option remember; floor((sqrt(1+8*n)-1)/2) end:
%p b:= proc(n) b(n):= (t-> binomial(t, n-t*(t+1)/2))(g(n)) end:
%p a:= n-> add(b(j)*b(n-j), j=0..n):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jun 04 2020
%t g[n_] := g[n] = Floor[(Sqrt[1 + 8*n] - 1)/2];
%t b[n_] := b[n] = Function[t, Binomial[t, n - t*(t + 1)/2]][g[n]];
%t a[n_] := Sum[b[j]*b[n - j], {j, 0, n}];
%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Mar 18 2022, after _Alois P. Heinz_ *)
%o (PARI) {a(n)=local(A=sum(m=0,(sqrt(8*n+1)+1)\2,x^(m*(m+1)/2)*(1+x+x*O(x^n))^m));polcoeff(A^2,n)} /* _Paul D. Hanna_, Apr 18 2012 */
%o for(n=0,66,print1(a(n),", "))
%Y Cf. A182152, A182153, A007318.
%K nonn
%O 0,2
%A _Philippe Deléham_, Nov 20 2008
%E More terms (and corrected term) from _L. Edson Jeffery_, Apr 23 2011
|
