A152037 Convolution of A007318 (Pascal's sequence) with itself .

1, 2, 3, 4, 7, 8, 9, 14, 20, 20, 21, 32, 43, 46, 51, 62, 71, 82, 107, 136, 145, 136, 144, 200, 280, 316, 296, 294, 359, 456, 535, 576, 591, 650, 820, 1020, 1078, 990, 963, 1160, 1541, 1950, 2225, 2244, 2034, 1892, 2211, 3024, 3866, 4260, 4207, 4066, 4150, 4630, 5617

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1035

Formula

G.f.: [Sum_{n>=0} x^(n*(n+1)/2) * (1+x)^n ]^2. [Paul D. Hanna, Apr 18 2012]

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]

Maple

g:= proc(n) option remember; floor((sqrt(1+8*n)-1)/2) end:
b:= proc(n) b(n):= (t-> binomial(t, n-t*(t+1)/2))(g(n)) end:
a:= n-> add(b(j)*b(n-j), j=0..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 04 2020

Mathematica

g[n_] := g[n] = Floor[(Sqrt[1 + 8*n] - 1)/2];
b[n_] := b[n] = Function[t, Binomial[t, n - t*(t + 1)/2]][g[n]];
a[n_] := Sum[b[j]*b[n - j], {j, 0, n}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

Prog

(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 */
for(n=0, 66, print1(a(n), ", "))

Crossrefs

Cf. A182152, A182153, A007318.

Sequence in context: A050271 A322742 A307561 * A329295 A058075 A243495

Adjacent sequences: A152034 A152035 A152036 * A152038 A152039 A152040

Keyword

nonn

Author

Philippe Deléham, Nov 20 2008

Extensions

More terms (and corrected term) from L. Edson Jeffery, Apr 23 2011

Status

approved