A000733 Boustrophedon transform of partition numbers 1, 1, 1, 2, 3, 5, 7, ...

1, 2, 4, 10, 30, 101, 394, 1760, 8970, 51368, 326991, 2289669, 17491625, 144760655, 1290204758, 12320541392, 125496010615, 1358185050788, 15563654383395, 188254471337718, 2396930376564860, 32044598671291610

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

Examples

			The array begins:
                   1
               1  ->   2
           4  <-   3  <-   1
       2  ->   6  ->   9  ->  10
  30  <-  28  <-  22  <-  13  <-   3
- _John Cerkan_, Jan 26 2017
		

Links

• John Cerkan, Table of n, a(n) for n = 0..482
• 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. A000041, A000751, A109449, A230957.

Programs

Haskell

a000733 n = sum $ zipWith (*) (a109449_row n) (1 : a000041_list)
-- Reinhard Zumkeller, Nov 04 2013

Mathematica

t[n_, 0] := If[n == 0, 1, PartitionsP[n-1]]; 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 count, accumulate, islice
from sympy import npartitions
def A000733_gen(): # generator of terms
    yield 1
    blist = (1,)
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist),initial=npartitions(i))))[-1]
A000733_list = list(islice(A000733_gen(),40)) # Chai Wah Wu, Jun 12 2022