A275872 A binomial convolution recurrence sequence.

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

Author

Olivier Gérard, Aug 11 2016

Keywords

• nonn
• eigen
• easy

Comments

Shifts 2 places left and decreases by one under a variant of binomial transform (see formula section).

Links

• Robert Israel, Table of n, a(n) for n = 0..648

Crossrefs

Cf. A000994, A007476, A032346.

Programs

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}]

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

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