A000674 Boustrophedon transform of 1, 2, 2, 2, 2, ...

1, 3, 7, 16, 43, 138, 527, 2346, 11943, 68418, 435547, 3050026, 23300443, 192835698, 1718682167, 16412205306, 167173350543, 1809239622978, 20732358910387, 250773962554186, 3192953259262243, 42686640718266258, 597853508941160207

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

Examples

			G.f. = 1 + 3*x + 7*x^2 + 16*x^3 + 43*x^4 + 138*x^5 + 527*x^6 + 2346*x^7 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..400
• Peter Luschny, An old operation on sequences: the Seidel transform.
• J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996), 44-54 (Abstract, pdf, ps).
• N. J. A. Sloane, Transforms.
• Wikipedia, Boustrophedon transform.
• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A000111, A040000, A109449.

Programs

Haskell

a000674 n = sum $ zipWith (*) (a109449_row n) (1 : repeat 2)
-- Reinhard Zumkeller, Nov 04 2013

Mathematica

With[{nn=30},CoefficientList[Series[(Sec[x]+Tan[x])(2Exp[x]-1),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 04 2015 *)

Python

from itertools import accumulate, islice
def A000674_gen(): # generator of terms
    yield 1
    blist = (1,)
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=2)))[-1]
A000674_list = list(islice(A000674_gen(),30)) # Chai Wah Wu, Jun 11 2022

Formula

a(n) = Sum_{k=0..n} A109449(n,k)*A040000(k). - Reinhard Zumkeller, Nov 04 2013

E.g.f.: (sec(x) + tan(x))*(2*exp(x) - 1). - Sergei N. Gladkovskii, Oct 28 2014

Binomial convolution of A000111 and A040000. - Michael Somos, Oct 30 2014

a(n) ~ n! * (2*exp(Pi/2)-1) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015

Extensions

More terms from Sean A. Irvine, Feb 20 2011