A338400 Inverse boustrophedon transform of the partition numbers.

1, 0, 1, -2, 2, -19, 39, -257, 1113, -6829, 42399, -299550, 2281531, -18901042, 168402645, -1608304966, 16381456532, -177291076953, 2031597803009, -24573784682206, 312883002507064, -4182938253898882, 58584703430964506, -857812167322107132, 13106404407407087063

Offset: 0

Author

Pontus von Brömssen, Oct 24 2020

Keywords

• sign

Links

• Eric Weisstein's World of Mathematics, Boustrophedon Transform
• Wikipedia, Boustrophedon transform
• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A000041, A000111, A000751.

Programs

Python

import sympy
def A338400(n):
  T=[]
  for k in range(n+1):
    T.append(sympy.npartitions(k))
    T.reverse()
    for i in range(k):
      T[i+1]=T[i]-T[i+1]
  return T[-1]

Python

from itertools import islice, count, accumulate
from operator import sub
from sympy import npartitions
def A338400_gen(): # generator of terms
    blist = tuple()
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist),func=sub,initial=npartitions(i))))[-1]
A338400_list = list(islice(A338400_gen(),20)) # Chai Wah Wu, Jun 10 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A000111(n-k) * A000041(k).