A000746 Boustrophedon transform of triangular numbers.

1, 4, 13, 39, 120, 407, 1578, 7042, 35840, 205253, 1306454, 9148392, 69887664, 578392583, 5155022894, 49226836114, 501420422112, 5426640606697, 62184720675718, 752172431553308, 9576956842743904, 128034481788227195

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. A000217, A000718, A109449.

Programs

Haskell

a000746 n = sum $ zipWith (*) (a109449_row n) $ tail a000217_list
-- Reinhard Zumkeller, Nov 03 2013

Magma

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( (Sec(x)+Tan(x))*Exp(x)*(x^2+4*x+2)/2 ))); // G. C. Greubel, Jul 10 2025

Mathematica

t[n_, 0] := (n + 1) (n + 2)/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 A000746_gen(): # generator of terms
    blist, c = tuple(), 1
    for i in count(2):
        yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]
        c += i
A000746_list = list(islice(A000746_gen(),40)) # Chai Wah Wu, Jun 12 2022

SageMath

@CachedFunction
def f(n, k):
    if (k==0): return binomial(n+2,2)
    else: return f(n, k-1) + f(n-1, n-k)
def A000746(n): return f(n,n)
[A000746(n) for n in range(31)] # G. C. Greubel, Jul 10 2025

Formula

a(n) = Sum_{k=0..n} A109449(n,k)*(k + 1)*(k + 2)/2. - Reinhard Zumkeller, Nov 03 2013

E.g.f.: (sec(x) + tan(x))*exp(x)*(x^2 + 4*x + 2)/2. - Sergei N. Gladkovskii, Oct 30 2014

a(n) ~ n! * (Pi^2 + 8*Pi + 8) * exp(Pi/2) * 2^(n-1) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015