A000732 Boustrophedon transform of 1 & primes: 1,2,3,5,7,...

1, 3, 8, 22, 66, 222, 862, 3838, 19542, 111894, 712282, 4987672, 38102844, 315339898, 2810523166, 26838510154, 273374835624, 2958608945772, 33903161435148, 410085034127000, 5221364826476796, 69804505809732988

Offset: 0

Author

N. J. A. Sloane and Simon Plouffe

Keywords

• nonn
• easy

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. A000747, A230953, A230954, A230955.

Programs

Haskell

a000732 n = sum $ zipWith (*) (a109449_row n) a008578_list

Mathematica

t[n_, 0] := If[n==0, 1, Prime[n]]; 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
from sympy import prime
def A000732_gen(): # generator of terms
    yield 1
    blist = (1,)
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=prime(i))))[-1]
A000732_list = list(islice(A000732_gen(),40)) # Chai Wah Wu, Jun 12 2022

Formula

a(n) = Sum_{k=0..n} A109449(n,k)*A008578(k+1). - Reinhard Zumkeller, Nov 04 2013

E.g.f.: (sec(x) + tan(x))*(1 + Sum_{k>=1} prime(k)*x^k/k!). - Ilya Gutkovskiy, Apr 23 2019