login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A306727 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) is the number of partitions of 3*n into powers of 3 less than or equal to 3^k. 1
1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 4, 1, 1, 2, 3, 5, 5, 1, 1, 2, 3, 5, 7, 6, 1, 1, 2, 3, 5, 7, 9, 7, 1, 1, 2, 3, 5, 7, 9, 12, 8, 1, 1, 2, 3, 5, 7, 9, 12, 15, 9, 1, 1, 2, 3, 5, 7, 9, 12, 15, 18, 10, 1, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 11, 1, 1, 2, 3, 5, 7, 9, 12, 15, 18, 23, 26, 12, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Column sequences converge to A005704.

LINKS

Serguei Zolotov, Table of n, a(n) for n = 0..10584

FORMULA

G.f. of column k: 1/(1-x) * 1/Product_{j=0..k-1} (1 - x^(3^j)).

EXAMPLE

A(3,3) = 5, because there are 5 partitions of 3*3=9 into powers of 3 less than or equal to 3^3=9: [9], [3,3,3], [3,3,1,1,1], [3,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1].

Square array A(n,k) begins:

  1, 1, 1, 1, 1, 1,  ...

  1, 2, 2, 2, 2, 2,  ...

  1, 3, 3, 3, 3, 3,  ...

  1, 4, 5, 5, 5, 5,  ...

  1, 5, 7, 7, 7, 7,  ...

  1, 6, 9, 9, 9, 9,  ...

MATHEMATICA

nmax = 12;

f[k_] := f[k] = 1/(1-x) 1/Product[1-x^(3^j), {j, 0, k-1}] + O[x]^(nmax+1) // CoefficientList[#, x]&;

A[n_, k_] := f[k][[n+1]];

Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-Fran├žois Alcover, Nov 20 2019 *)

PROG

(Python)

def aseq(p, x, k):

    # generic algorithm for any p - power base, p=3 for this sequence

    if x < 0:

        return 0

    if x < p:

        return 1

    # coefficients

    arr = [0]*(x+1)

    arr[0] = 1

    m = p**k

    while m > 0:

        for i in range(m, x+1, m):

            arr[i] += arr[i-m]

        m //= p

    return arr[x]

def A(n, k):

    p = 3

    return aseq(p, p*n, k)

# A(n, k), 5 = A(3, 3) = aseq(3, 3*3, 3)

# Serguei Zolotov, Mar 13 2019

CROSSREFS

Main diagonal gives A005704.

A181322 gives array for base p=2.

Sequence in context: A176484 A144328 A128227 * A324209 A228107 A140207

Adjacent sequences:  A306724 A306725 A306726 * A306728 A306729 A306730

KEYWORD

nonn,tabl

AUTHOR

Serguei Zolotov, Mar 06 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 4 07:10 EST 2021. Contains 341781 sequences. (Running on oeis4.)