OFFSET
0,3
COMMENTS
The self-convolution equals A051163. - Paul D. Hanna, Nov 23 2004
Equals row sums of triangle A152193. - Gary W. Adamson, Nov 28 2008
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
FORMULA
G.f. A(x) satisfies A(x^2) = A(x/(1+x))/(1+x) and A(x) = A(x^2/(1-x)^2)/(1-x).
The recursive formula A[n+1] = A[n](x^2/(1-x)^2)/(1-x), A[0]=1, yields exactly 2^n terms after n iterations: A(x) - A[n](x) = x^(2^n) + (2^n+1)*x^(2^n+1) + O(x^(2^n+2)). For example, A[4] = (1-x)^3*(1-2*x-x^2)/((1-2*x)(1-4*x+4*x^2-2*x^4)) = A(x) - x^16 - 17*x^17 + O(x^18). - M. F. Hasler, Aug 19 2015
E.g.f.: exp(x) * Sum_{n>=0} a(n) * x^(2*n) / (2*n)!. - Ilya Gutkovskiy, Feb 26 2022
The expansion of exp(Sum_{n >= 1} a(n)*(2*x)^n/n!) = 1 + 2*x + 6*x^2 + 20*x^3 + 74*x^4 + 292*x^5 + 1204*x^6 + ... appears to have integer coefficients. Equivalently, the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for positive integers k and n and all primes p >= 3. - Peter Bala, Jan 11 2023
EXAMPLE
Array of successive differences (col. 1 is the inverse binomial transform):
1, 1, 2, 4, 9, 21, 50, ...
0, 1, 2, 5, 12, 29, 70, ...
1, 1, 3, 7, 17, 41, ...
0, 2, 4, 10, 24, 59, ...
2, 2, 6, 14, 35, 87, ...
0, 4, 8, 21, 52, ...
4, 4, 13, 31, 79, ...
0, 9, 18, 48, ...
9, 9, 30, ...
...
MAPLE
a:= proc(n) option remember; add(`if`(k=0, 1,
`if`(k::odd, 0, a(k/2)))*binomial(n, k), k=0..n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Jul 08 2015
MATHEMATICA
a[n_] := a[n] = Sum[If[k == 0, 1, If[OddQ[k], 0, a[k/2]]]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 20 2017, translated from Maple *)
PROG
(PARI) a(n)=local(A, m); if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=2; A=subst(A, x, (x/(1-x))^2)/(1-x)); polcoeff(A, n))
(PARI) a=List(); for(n=1, 100, listput(a, sum(i=1, n\2, a[i]*binomial(n, 2*i), 1))) \\ M. F. Hasler, Aug 19 2015
CROSSREFS
KEYWORD
nonn,eigen
AUTHOR
EXTENSIONS
Incorrect g.f. and formulas removed by R. J. Mathar, Oct 02 2012
Incorrect g.f.s removed by Peter Bala, Jul 07 2015
STATUS
approved