A286708 Powerful numbers (A001694) that are not prime powers (A000961).

36, 72, 100, 108, 144, 196, 200, 216, 225, 288, 324, 392, 400, 432, 441, 484, 500, 576, 648, 675, 676, 784, 800, 864, 900, 968, 972, 1000, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2025, 2116, 2304, 2312, 2500, 2592, 2601, 2700, 2704, 2744

Offset

1,1

Comments

If a prime p divides a(n) then p^2 must also divide a(n) and number of distinct primes dividing a(n) > 1.

Intersection of A001694 and A024619.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 5997 terms from Robert Israel)

Eric Weisstein's World of Mathematics, Prime Power.

Eric Weisstein's World of Mathematics, Powerful Number.

Index entries for sequences related to powerful numbers

Formula

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{p prime} 1/(p*(p-1)) - 1 = A082695 - A136141 - 1 = 0.17043976777096407719... - Amiram Eldar, Feb 12 2021

Example

-------------------------------
| n | a(n) | prime            |
|   |      | factorization    |
|------------------------------
| 1 | 36   | {{2, 2}, {3, 2}} |
| 2 | 72   | {{2, 3}, {3, 2}} |
| 3 | 100  | {{2, 2}, {5, 2}} |
| 4 | 108  | {{2, 2}, {3, 3}} |
| 5 | 144  | {{2, 4}, {3, 2}} |
| 6 | 196  | {{2, 2}, {7, 2}} |
| 7 | 200  | {{2, 3}, {5, 2}} |
| 8 | 216  | {{2, 3}, {3, 3}} |
| 9 | 225  | {{3, 2}, {5, 2}} |
-------------------------------
a(n) = p_1^e_1*p_2^e_2*... : {{p_1, e_1}, {p_2, e_2}, ...}.

Maple

N:= 10000:
S:= {1}: P:= {1}:
p:= 1:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  S:= map(s -> (s, seq(s*p^k, k = 2 .. floor(log[p](N/s)))), S);
  P:= P union {seq(p^k, k=2..floor(log[p](N)))}:
od:
sort(convert(S minus P, list)); # Robert Israel, May 14 2017

Mathematica

Select[Range@2750, Min@FactorInteger[#][[All, 2]] > 1 && ! PrimePowerQ[#] &]
(* Second program *)
nn = 2^25; Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], ! PrimePowerQ[#] &] (* Michael De Vlieger, Jun 22 2022 *)

Prog

(Python)
from sympy import primefactors, factorint
print([n for n in range(4, 2745) if len(primefactors(n)) > 1 and min(list(factorint(n).values())) > 1]) # Karl-Heinz Hofmann, Feb 07 2023
(Python)
from math import isqrt
from sympy import integer_nthroot, primepi, mobius
def A286708(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x, 3)[0])-l
        return c+1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f, n, n) # Chai Wah Wu, Sep 10 2024

Crossrefs

Cf. A000961, A001221, A001694, A024619, A082695, A136141, A131605.

Keyword

nonn

Author

_Ilya Gutkovskiy_, May 13 2017

Status

approved

A304038 Irregular triangle T(n,k) read by rows: first row is 0, n-th row (n > 1) lists indices of distinct primes dividing n.

0, 1, 2, 1, 3, 1, 2, 4, 1, 2, 1, 3, 5, 1, 2, 6, 1, 4, 2, 3, 1, 7, 1, 2, 8, 1, 3, 2, 4, 1, 5, 9, 1, 2, 3, 1, 6, 2, 1, 4, 10, 1, 2, 3, 11, 1, 2, 5, 1, 7, 3, 4, 1, 2, 12, 1, 8, 2, 6, 1, 3, 13, 1, 2, 4, 14, 1, 5, 2, 3, 1, 9, 15, 1, 2, 4, 1, 3, 2, 7, 1, 6, 16, 1, 2, 3, 5, 1, 4, 2, 8, 1, 10, 17, 1, 2, 3, 18, 1, 11

Offset

1,3

Links

Table of n, a(n) for n=1..100.

Eric Weisstein's World of Mathematics, Distinct Prime Factors

Index entries for sequences computed from indices in prime factorization

Formula

T(n,k) = A000720(A027748(n,k)).

T(n,1) = A055396(n).

T(n,A001221(n)) = A061395(n).

Example

The irregular triangle begins:
1:  {0}
2:  {1}
3:  {2}
4:  {1}
5:  {3}
6:  {1, 2}
7:  {4}
8:  {1}
9:  {2}
10: {1, 3}
11: {5}
12: {1, 2}

Mathematica

Flatten[Table[PrimePi[FactorInteger[n][[All, 1]]], {n, 1, 62}]]

Crossrefs

Cf. A000040, A000720, A001221 (row lengths), A027748, A055396, A061395, A066328 (row sums), A112798, A156061 (row products), A302170.

Keyword

nonn,tabf

Author

_Ilya Gutkovskiy_, May 05 2018

Status

approved

A304969 Expansion of 1/(1 - Sum_{k>=1} q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

1, 1, 2, 5, 11, 25, 57, 129, 292, 662, 1500, 3398, 7699, 17443, 39519, 89536, 202855, 459593, 1041267, 2359122, 5344889, 12109524, 27435660, 62158961, 140828999, 319065932, 722884274, 1637785870, 3710611298, 8406859805, 19046805534, 43152950024, 97768473163

Offset

0,3

Comments

Invert transform of A000009.

From Gus Wiseman, Jul 31 2022: (Start)

Also the number of ways to choose a multiset partition into distinct constant multisets of a multiset of length n that covers an initial interval of positive integers. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 11 multiset partitions are:

{{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}

{{1},{2}} {{1},{1,1}} {{1},{1,1,1}}

{{1},{2,2}} {{1,1},{2,2}}

{{2},{1,1}} {{1},{2,2,2}}

{{1},{2},{3}} {{2},{1,1,1}}

{{1},{2},{1,1}}

{{1},{2},{2,2}}

{{1},{2},{3,3}}

{{1},{3},{2,2}}

{{2},{3},{1,1}}

{{1},{2},{3},{4}}

The non-strict version is A055887.

The strongly normal non-strict version is A063834.

The strongly normal version is A270995.

(End)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2816

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Partition Function Q

Index entries for sequences related to partitions

Index entries for sequences related to compositions

Formula

G.f.: 1/(1 - Sum_{k>=1} A000009(k)*x^k).

G.f.: 1/(2 - Product_{k>=1} (1 + x^k)).

G.f.: 1/(2 - Product_{k>=1} 1/(1 - x^(2*k-1))).

G.f.: 1/(2 - exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)))).

a(n) ~ c / r^n, where r = 0.441378990861652015438479635503868737167721352874... is the root of the equation QPochhammer[-1, r] = 4 and c = 0.4208931614610039677452560636348863586180784719323982664940444607322... - Vaclav Kotesovec, May 23 2018

Example

From Gus Wiseman, Jul 31 2022: (Start)
a(n) is the number of ways to choose a strict integer partition of each part of an integer composition of n. The a(1) = 1 through a(4) = 11 choices are:
  ((1))  ((2))     ((3))        ((4))
         ((1)(1))  ((21))       ((31))
                   ((1)(2))     ((1)(3))
                   ((2)(1))     ((2)(2))
                   ((1)(1)(1))  ((3)(1))
                                ((1)(21))
                                ((21)(1))
                                ((1)(1)(2))
                                ((1)(2)(1))
                                ((2)(1)(1))
                                ((1)(1)(1)(1))
(End)

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(j)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

Mathematica

nmax = 32; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - 1/QPochhammer[x, x^2]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

Crossrefs

Cf. A050342, A055887, A067687, A081362, A279785, A299106.

Row sums of A308680.

The unordered version is A089259, non-strict A001970 (row-sums of A061260).

For partitions instead of compositions we have A270995, non-strict A063834.

A000041 counts integer partitions, strict A000009.

A072233 counts partitions by sum and length.

Cf. A279784.

Keyword

nonn

Author

_Ilya Gutkovskiy_, May 22 2018

Status

approved

A333806 Number of distinct prime divisors of n that are < sqrt(n).

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 3, 0, 1, 2, 1, 1, 2, 0, 1, 1, 3, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 3

Offset

1,12

Comments

a(n) = 0 if and only if n = p^k where p is prime and k is 0, 1, or 2. - Charles R Greathouse IV, Apr 07 2020

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

G.f.: Sum_{k>=1} x^(prime(k)*(prime(k) + 1)) / (1 - x^prime(k)).

a(k*n) >= a(n) for k > 0. a(n^e) = A001221(n) for e > 2. - Richard Peterson, Dec 19 2024

Maple

N:= 100: # for a(1)..a(N)
V:= Vector(N):
p:= 1:
do
  p:= nextprime(p);
  if p^2 >= N then break fi;
  L:= [seq(p*k, k=p+1..N/p)];
  V[L]:= V[L]+~1
od:
convert(V, list); # Robert Israel, Apr 07 2020

Mathematica

Table[DivisorSum[n, 1 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Prog

(PARI) a(n)=my(f=factor(n)[, 1]); sum(i=1, #f, f[i]^2<n) \\ Charles R Greathouse IV, Apr 07 2020

Crossrefs

Cf. A001221, A056924, A063962, A333805, A333808.

Keyword

nonn

Author

_Ilya Gutkovskiy_, Apr 05 2020

Status

approved

A304961 Expansion of Product_{k>=1} (1 + 2^(k-1)*x^k).

1, 1, 2, 6, 12, 32, 72, 176, 384, 960, 2112, 4992, 11264, 26112, 58368, 136192, 301056, 688128, 1548288, 3489792, 7766016, 17596416, 38993920, 87293952, 194248704, 432537600, 957349888, 2132803584, 4699717632, 10406068224, 23001563136, 50683969536, 111434268672, 245819768832

Offset

0,3

Comments

Number of compositions of partitions of n into distinct parts. a(3) = 6: 3, 21, 12, 111, 2|1, 11|1. - Alois P. Heinz, Sep 16 2019

Also the number of ways to split a composition of n into contiguous subsequences with strictly decreasing sums. - Gus Wiseman, Jul 13 2020

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = (-1) * 2^(n-1). - Seiichi Manyama, Aug 22 2020

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for sequences related to partitions

Index entries for sequences related to compositions

Formula

G.f.: Product_{k>=1} (1 + A011782(k)*x^k).

a(n) ~ 2^n * exp(2*sqrt(-polylog(2, -1/2)*n)) * (-polylog(2, -1/2))^(1/4) / (sqrt(6*Pi) * n^(3/4)). - Vaclav Kotesovec, Sep 19 2019

Example

From Gus Wiseman, Jul 13 2020: (Start)
The a(0) = 1 through a(4) = 12 splittings:
  ()  (1)  (2)    (3)        (4)
           (1,1)  (1,2)      (1,3)
                  (2,1)      (2,2)
                  (1,1,1)    (3,1)
                  (2),(1)    (1,1,2)
                  (1,1),(1)  (1,2,1)
                             (2,1,1)
                             (3),(1)
                             (1,1,1,1)
                             (1,2),(1)
                             (2,1),(1)
                             (1,1,1),(1)
(End)

Mathematica

nmax = 33; CoefficientList[Series[Product[(1 + 2^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Prog

(PARI) N=40; x='x+O('x^N); Vec(prod(k=1, N, 1+2^(k-1)*x^k)) \\ Seiichi Manyama, Aug 22 2020

Crossrefs

Cf. A000009, A011782, A022629, A098407, A102866, A266964, A271619, A279785.

The non-strict version is A075900.

Starting with a reversed partition gives A323583.

Starting with a partition gives A336134.

Partitions of partitions are A001970.

Splittings with equal sums are A074854.

Splittings of compositions are A133494.

Splittings with distinct sums are A336127.

Cf. A006951, A063834, A316245, A317715, A319794, A323582, A336135, A336136.

Keyword

nonn

Author

_Ilya Gutkovskiy_, May 22 2018

Status

approved

A333805 Number of odd divisors of n that are < sqrt(n).

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 4, 2, 1, 2, 1, 2, 2, 1, 2, 3, 2, 1, 2, 1, 1, 4

Offset

1,12

Comments

If we define a divisor d|n to be strictly inferior if d < n/d, then strictly inferior divisors are counted by A056924 and listed by A341674. This sequence counts strictly inferior odd divisors. - Gus Wiseman, Feb 26 2021

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

G.f.: Sum_{k>=1} x^(2*k*(2*k - 1)) / (1 - x^(2*k - 1)).

Example

The strictly inferior odd divisors of 945 are 1, 3, 5, 7, 9, 15, 21, 27, so a(945) = 8. - Gus Wiseman, Feb 27 2021

Mathematica

Table[DivisorSum[n, 1 &, # < Sqrt[n] && OddQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[x^(2 k (2 k - 1))/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Prog

(PARI) A333805(n) = sumdiv(n, d, (d%2)&&((d*d)<n)); \\ Antti Karttunen, Nov 02 2022

Crossrefs

Dominated by A001227 (number of odd divisors).

Strictly inferior divisors (not just odd) are counted by A056924.

The non-strict version is A069288.

These divisors add up to A070039.

The case of prime divisors is A333806.

The strictly superior version is A341594.

The case of squarefree divisors is A341596.

The superior version is A341675.

The case of prime-power divisors is A341677.

A006530 selects the greatest prime factor.

A020639 selects the smallest prime factor.

- Odd -

A000009 counts partitions into odd parts, ranked by A066208.

A026424 lists numbers with odd Omega.

A027193 counts odd-length partitions.

A067659 counts strict partitions of odd length, ranked by A030059.

- Inferior divisors -

A033676 selects the greatest inferior divisor.

A033677 selects the smallest superior divisor.

A038548 counts superior (or inferior) divisors.

A060775 selects the greatest strictly inferior divisor.

A341674 lists strictly inferior divisors.

Cf. A001248, A051283, A063538/A063539, A063962, A116883/A116882, A300272, A333807, A333809, A340832.

Keyword

nonn

Author

_Ilya Gutkovskiy_, Apr 05 2020

Extensions

Data section extended up to a(105) by Antti Karttunen, Nov 02 2022

Status

approved

A286335 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^k.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 2, 0, 1, 5, 10, 13, 9, 3, 0, 1, 6, 15, 24, 24, 14, 4, 0, 1, 7, 21, 40, 51, 42, 22, 5, 0, 1, 8, 28, 62, 95, 100, 73, 32, 6, 0, 1, 9, 36, 91, 162, 206, 190, 120, 46, 8, 0, 1, 10, 45, 128, 259, 384, 425, 344, 192, 66, 10, 0

Offset

0,8

Comments

A(n,k) is the number of partitions of n into distinct parts (or odd parts) with k types of each part.

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

N. J. A. Sloane, Transforms

Formula

G.f. of column k: Product_{j>=1} (1 + x^j)^k.

A(n,k) = Sum_{i=0..k} binomial(k,i) * A308680(n,k-i). - Alois P. Heinz, Aug 29 2019

Example

A(3,2) = 6 because we have [3], [3'], [2, 1], [2', 1], [2, 1'] and [2', 1'] (partitions of 3 into distinct parts with 2 types of each part).
Also A(3,2) = 6 because we have [3], [3'], [1, 1, 1], [1, 1, 1'], [1, 1', 1'] and [1', 1', 1'] (partitions of 3 into odd parts with 2 types of each part).
Square array begins:
  1,  1,  1,   1,   1,   1,  ...
  0,  1,  2,   3,   4,   5,  ...
  0,  1,  3,   6,  10,  15,  ...
  0,  2,  6,  13,  24,  40,  ...
  0,  2,  9,  24,  51,  95,  ...
  0,  3, 14,  42, 100, 206,  ...

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..n/i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 29 2019

Mathematica

Table[Function[k, SeriesCoefficient[Product[(1 + x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Crossrefs

Columns k=0-32 give: A000007, A000009, A022567-A022596.

Rows n=0-2 give: A000012, A001477, A000217.

Main diagonal gives A270913.

Antidiagonal sums give A299106.

Cf. A144064, A286352, A308680.

Keyword

nonn,tabl

Author

_Ilya Gutkovskiy_, May 07 2017

Status

approved

A318029 Expansion of Sum_{k>=2} x^(k*(k+3)/2) / Product_{j=1..k} (1 - x^j).

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 9, 11, 14, 16, 20, 24, 28, 34, 40, 47, 55, 65, 75, 88, 102, 118, 136, 158, 180, 208, 238, 272, 311, 355, 403, 459, 521, 590, 668, 756, 852, 962, 1084, 1218, 1370, 1538, 1724, 1932, 2163, 2417, 2701, 3015, 3361, 3745, 4170, 4636, 5154, 5724

Offset

0,8

Comments

Number of partitions of n into at least two distinct parts >= 2.

Links

Table of n, a(n) for n=0..60.

Index entries for sequences related to partitions

Formula

G.f.: x - 1/(1 - x) + Product_{k>=2} (1 + x^k).

a(n) = A025147(n) - 1 for n > 1.

Example

a(9) = 4 because we have [7, 2], [6, 3], [5, 4] and [4, 3, 2].

Mathematica

nmax = 60; CoefficientList[Series[Sum[x^(k (k + 3)/2)/Product[(1 - x^j), {j, 1, k}], {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[x - 1/(1 - x) + 1/((1 + x) QPochhammer[x, x^2]), {x, 0, nmax}], x]
Join[{0, 0}, Table[-1 + Sum[(-1)^(n - k) PartitionsQ[k], {k, 0, n}], {n, 2, 60}]]

Crossrefs

Cf. A000009, A025147, A083751, A096765, A111133.

Keyword

nonn

Author

_Ilya Gutkovskiy_, Aug 13 2018

Status

approved

A297328 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^k.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 6, 0, 1, 4, 12, 18, 14, 0, 1, 5, 18, 37, 49, 25, 0, 1, 6, 25, 64, 114, 114, 56, 0, 1, 7, 33, 100, 219, 312, 282, 97, 0, 1, 8, 42, 146, 375, 676, 855, 624, 198, 0, 1, 9, 52, 203, 594, 1276, 2030, 2178, 1422, 354, 0, 1, 10, 63, 272, 889, 2196, 4155, 5736, 5496, 3058, 672, 0

Offset

0,8

Links

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Formula

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

A(0,k) = 1; A(n,k) = (k/n) * Sum_{j=1..n} A078308(j) * A(n-j,k). - Seiichi Manyama, Aug 16 2023

Example

G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k + 5)*x^2 + (1/6)*k*(k^2 + 15*k + 20)*x^3 + (1/24)*k*(k^3 + 30*k^2 + 155*k + 150)*x^4 + (1/120)*k*(k^4 + 50*k^3 + 575*k^2 + 1750*k + 624)*x^5 + ...
Square array begins:
  1,   1,    1,    1,    1,     1,  ...
  0,   1,    2,    3,    4,     5,  ...
  0,   3,    7,   12,   18,    25,  ...
  0,   6,   18,   37,   64,   100,  ...
  0,  14,   49,  114,  219,   375,  ...
  0,  25,  114,  312,  676,  1276,  ...

Mathematica

Table[Function[k, SeriesCoefficient[Product[1/(1 - i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Prog

(PARI) first(n, k) = my(res = matrix(n, k)); for(u=1, k, my(col = Vec(prod(j=1, n, 1/(1 - j*x^j)^(u-1)) + O(x^n))); for(v=1, n, res[v, u] = col[v])); res \\ Iain Fox, Dec 28 2017

Crossrefs

Columns k=0..32 give A000007, A006906, A022726, A022727, A022728, A022729, A022730, A022731, A022732, A022733, A022734, A022735, A022736, A022737, A022738, A022739, A022740, A022741, A022742, A022743, A022744, A022745, A022746, A022747, A022748, A022749, A022750, A022751, A022752, A022753, A022754, A022755, A022756.

Main diagonal gives A297329.

Antidiagonal sums give A299162.

Cf. A078308, A266941, A297321, A297323, A297325.

Keyword

nonn,tabl

Author

_Ilya Gutkovskiy_, Dec 28 2017

Status

approved

A341243 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^4.

1, 0, 4, 4, 10, 16, 26, 44, 63, 100, 144, 212, 297, 420, 584, 796, 1081, 1452, 1940, 2556, 3355, 4372, 5668, 7288, 9327, 11892, 15076, 19012, 23884, 29904, 37276, 46284, 57276, 70680, 86918, 106528, 130220, 158784, 193054, 234076, 283178, 341824, 411616, 494512, 592933

Offset

4,3

Links

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Formula

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^4.

a(n) ~ A112160(n). - Vaclav Kotesovec, Feb 20 2021

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..48);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax = 48; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

Crossrefs

Cf. A000700, A001482, A022599, A112160, A327382, A338463, A341222, A341241, A341244, A341245, A341246, A341247, A341251.

Column k=4 of A341279.

Keyword

nonn

Author

_Ilya Gutkovskiy_, Feb 07 2021

Status

approved