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A275872
A binomial convolution recurrence sequence.
1
0, 0, 1, 1, 2, 6, 18, 54, 173, 605, 2274, 9020, 37486, 163128, 743101, 3535765, 17518018, 90126158, 480514430, 2650912738, 15112253425, 88903779401, 539003066674, 3363608949132, 21581457167994, 142227480847092, 961868098767105, 6669657795455817, 47380035801732034, 344555811578909254, 2563218995058696890
OFFSET
0,5
COMMENTS
Shifts 2 places left and decreases by one under a variant of binomial transform (see formula section).
LINKS
FORMULA
Sum_{i=0..n} binomial(n+1,i+1)*a(i) = a(n+2) - 1.
G.f. g(x) satisfies g(x) = x^2/(1-x) + x^2*g(x/(1-x))/(1-x)^2. - Robert Israel, Aug 28 2016
MAPLE
A[0]:= 0:
A[1]:= 0:
for m from 2 to 50 do
A[m]:= 1 + add(binomial(m-1, i+1)*A[i], i=0..m-2)
od:
seq(A[i], i=0..50); # Robert Israel, Aug 28 2016
MATHEMATICA
Clear[a]; a[0] = 0 ; a[1] = 0; a[n_] := a[n] = 1 + Sum[Binomial[n - 1, j+1]*a[j], {j, 0, n - 1}]; Table[a[n], {n, 0, 22}]
PROG
(PARI) first(n)=my(v=vector(n)); for(k=0, n-2, v[k+2]=sum(i=2, k, binomial(k+1, i+1)*v[i])+1); concat(0, v) \\ Charles R Greathouse IV, Aug 29 2016
CROSSREFS
KEYWORD
nonn,eigen,easy
AUTHOR
Olivier Gérard, Aug 11 2016
STATUS
approved