A195442 G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1+k*x)^4.

1, 4, 38, 604, 13797, 416168, 15711490, 715565508, 38278167466, 2356801113088, 164391501446500, 12826066272723832, 1107734575497955773, 104985517080964044744, 10838759916371136482170, 1211294776167588947134004, 145737426974188872882131694

Offset

0,2

Comments

CONJECTURES. (Start)

_ The self-convolution square-root is A195443 and consists entirely of integers.

_ a(4*(2^n-1)) is odd; odd numbers occur only at positions {2^(n+2) - 4, n>=0}.

_ In the self-convolution square-root of this sequence (A195443), odd numbers occur only at positions {2^(n+1) - 2, n>=0}. (End)

Links

Paul D. Hanna, Table of n, a(n) for n = 0..199

Example

G.f.: 1 = 1/(1+x)^4 + 4*x/((1+x)^4*(1+2*x)^4) + 38*x^2/((1+x)^4*(1+2*x)^4*(1+3*x)^4) + 604*x^3/((1+x)^4*(1+2*x)^4*(1+3*x)^4*(1+4*x)^4) +...
This sequence has odd terms at [0,4,12,28,60,124,...,2^(n+2)-4,...].
O.g.f.: A(x) = 1 + 4*x + 38*x^2 + 604*x^3 + 13797*x^4 + 416168*x^5 +...
where the square-root is an integer series (cf. A195443):
A(x)^(1/2) = 1 + 2*x + 17*x^2 + 268*x^3 + 6218*x^4 + 191092*x^5 + 7331943*x^6 + 338203880*x^7 + 18267488524*x^8 + 1132962942756*x^9 +...
which has odd terms at [0,2,6,14,30,62,126,...,2^(n+1)-2,...].

Mathematica

a[n_] := a[n] = If[n == 0, 1, SeriesCoefficient[1-Sum[a[k] x^k/Product[1 + j x + x O[x]^n, {j, 1, k+1}]^4, {k, 0, n-1}], {x, 0, n}]];
a /@ Range[0, 16] (* Jean-François Alcover, Nov 03 2019, from PARI *)

Prog

(PARI) {a(n)=if(n==0, 1, polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, 1+j*x+x*O(x^n))^4), n))}

Crossrefs

Cf. A118804, A193333, A195443.

Sequence in context: A364816 A277869 A138214 * A138562 A354686 A177382

Adjacent sequences: A195439 A195440 A195441 * A195443 A195444 A195445

Keyword

nonn

Author

Paul D. Hanna, Sep 18 2011

Status

approved