A000745 Boustrophedon transform of squares.

1, 5, 18, 57, 180, 617, 2400, 10717, 54544, 312353, 1988104, 13921501, 106350816, 880162337, 7844596536, 74910367309, 763030711936, 8257927397569, 94628877364936, 1144609672707741, 14573622985067744, 194834987492011649, 2728787718495477144, 39955604972310966797

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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. A000290, A000697.

Programs

Haskell

a000745 n = sum $ zipWith (*) (a109449_row n) $ tail a000290_list
-- Reinhard Zumkeller, Nov 03 2013

Mathematica

CoefficientList[Series[E^(x)*(x^2+3*x+1)*(1+Sin[x])/Cos[x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 12 2015 *)
t[n_, 0] := (n + 1)^2; t[n_, k_] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

Python

from itertools import accumulate, count, islice
def A000745_gen(): # generator of terms
    blist, c = tuple(), 1
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]
        c += 2*i+1
A000745_list = list(islice(A000745_gen(),40)) # Chai Wah Wu, Jun 12 2022

Formula

a(n) ~ n! * (6 + Pi + 4/Pi) * exp(Pi/2) * 2^n / Pi^n. - Vaclav Kotesovec, Jun 12 2015

E.g.f.: exp(x)*(x^2 + 3*x + 1)*(1+sin(x))/cos(x). - Vaclav Kotesovec, Jun 12 2015