A121870 Monthly Problem 10791, second expression.

1, 1, 2, 9, 61, 554, 6565, 96677, 1716730, 36072181, 881242577, 24674241834, 783024550969, 27896201305769, 1106485798248706, 48517267642373105, 2337333266369553253, 123040664089658462650, 7043260281573138384701, 436533086101058798529933

Offset: 0

Author

N. J. A. Sloane, Sep 05 2006

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..325
• A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.

Crossrefs

Cf. A024429, A024430, A121867, A121868, A121869.

Programs

GAP

List([0..25], n-> (Sum([0..Int(n/2)], k-> Stirling2(n,2*k)*(-1)^(k)) )^2 + (Sum([0..Int(n/2)], k-> (-1)^k*Stirling2(n,2*k+1)))^2 ); # G. C. Greubel, Oct 08 2019

Magma

C:= ComplexField(); a:= func< n | Round(Abs( (&+[I^k*StirlingSecond(n,k): k in [0..n]])^2 )) >;
[a(n): n in [0..25]]; // G. C. Greubel, Oct 08 2019

Maple

A121870a:= proc(a) local i, t:
i:=1: t:=0: for i to 100 do t:=t+evalf((i^(a-1))*(I)^i/(i)!): od:
RETURN(round(abs(t^2))):
end: a:= A121870a(n);
# Russell Walsmith, Apr 18 2008
# Alternate:
seq(abs(BellB(n,I))^2, n=0..30); # Robert Israel, Oct 15 2017

Mathematica

Table[Abs[BellB[n, I]]^2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2017 *)

PARI

a(n) = abs( (sum(k=0,n, I^k*stirling(n,k,2)))^2 );
vector(25, n, a(n-1)) \\ G. C. Greubel, Oct 08 2019

Sage

[abs( sum(I^k*stirling_number2(n,k) for k in (0..n))^2 ) for n in (0..25)] # G. C. Greubel, Oct 08 2019

Formula

a(n) = A121867(n)^2 + A121868(n)^2.

From Gary W. Adamson, Jul 22 2011: (Start)

sqrt(a(n)) = upper left term in M^n as to the modulus of a polar term; M = an infinite square production matrix in which a column of (i, i, i, ...) is appended to the right of Pascal's triangle, as follows (with i = sqrt(-1)):

1, i, 0, 0, 0, ...

1, 1, i, 0, 0, ...

1, 2, 1, i, 0, ...

1, 3, 3, 1, i, ...

... (End)

a(n) = |B_n(i)|^2, where B_n(x) is the n-th Bell polynomial, i = sqrt(-1) is the imaginary unit. - Vladimir Reshetnikov, Oct 15 2017

a(n) ~ (n*exp(-1 + Re(LambertW(i*n)) / Abs(LambertW(i*n))^2) / Abs(LambertW(i*n)))^(2*n) / Abs(1 + LambertW(i*n)), where i is the imaginary unit. - Vaclav Kotesovec, Jul 28 2021