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A365526
a(n) = Sum_{k=0..floor((n-1)/4)} Stirling2(n,4*k+1).
3
0, 1, 1, 1, 1, 2, 16, 141, 1051, 6953, 42571, 247886, 1401676, 7868005, 45210257, 277899961, 1917140421, 15186484134, 135259346092, 1295096363273, 12821558136891, 128268683204737, 1283599391456735, 12817818177339530, 127998022119881272
OFFSET
0,6
LINKS
FORMULA
Let A(0)=1, B(0)=0, C(0)=0 and D(0)=0. Let B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k), C(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), D(n+1) = Sum_{k=0..n} binomial(n,k)*C(k) and A(n+1) = Sum_{k=0..n} binomial(n,k)*D(k). A365525(n) = A(n), a(n) = B(n), A365527(n) = C(n) and A099948(n) = D(n).
G.f.: Sum_{k>=0} x^(4*k+1) / Product_{j=1..4*k+1} (1-j*x).
MAPLE
f:= proc(n) local k; add(Stirling2(n, 4*k+1), k=0..(n-1)/4) end proc:
map(f, [$0..30]); # Robert Israel, Sep 11 2024
MATHEMATICA
a[n_] := Sum[StirlingS2[n, 4*k+1], {k, 0, Floor[(n-1)/4]}]; Array[a, 25, 0] (* Amiram Eldar, Sep 13 2023 *)
PROG
(PARI) a(n) = sum(k=0, (n-1)\4, stirling(n, 4*k+1, 2));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 08 2023
STATUS
approved