A000751 Boustrophedon transform of partition numbers.

1, 2, 5, 14, 42, 143, 555, 2485, 12649, 72463, 461207, 3229622, 24671899, 204185616, 1819837153, 17378165240, 177012514388, 1915724368181, 21952583954117, 265533531724484, 3380877926676504, 45199008472762756

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			The array begins:
                   1
               1  ->   2
           5  <-   4  <-   2
       3  ->   8  ->  12  ->  14
  42  <-  39  <-  31  <-  19  <-   5
- _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. A000733, A230957.

Programs

Haskell

a000751 n = sum $ zipWith (*) (a109449_row n) a000041_list
-- Reinhard Zumkeller, Nov 03 2013

Mathematica

t[n_, 0] := PartitionsP[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 npartitions
def A000751_gen(): # generator of terms
    blist = tuple()
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist),initial=npartitions(i))))[-1]
A000751_list = list(islice(A000751_gen(),40)) # Chai Wah Wu, Jun 12 2022

Formula

a(n) = Sum_{k=0..n} A109449(n,k)*A000041(k). - Reinhard Zumkeller, Nov 03 2013