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

0,3

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.

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

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 *)

Prog

(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
(Magma) C<I>:= 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
(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
(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

Crossrefs

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

Sequence in context: A107883 A088182 A006155 * A268450 A146887 A173498

Adjacent sequences: A121867 A121868 A121869 * A121871 A121872 A121873

Keyword

nonn

Author

N. J. A. Sloane, Sep 05 2006

Status

approved