A341243 -id:A341243 - cp's OEIS frontend

A341279 Triangle read by rows: T(n,k) = coefficient of x^n in expansion of (-1 + Product_{j>=1} 1 / (1 + (-x)^j))^k, n >= 0, 0 <= k <= n.

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 3, 3, 4, 0, 1, 0, 1, 4, 6, 4, 5, 0, 1, 0, 2, 5, 9, 10, 5, 6, 0, 1, 0, 2, 8, 13, 16, 15, 6, 7, 0, 1, 0, 2, 9, 21, 26, 25, 21, 7, 8, 0, 1, 0, 2, 12, 27, 44, 45, 36, 28, 8, 9, 0, 1, 0, 3, 15, 40, 63, 80, 71, 49, 36, 9, 10, 0, 1

Offset: 0

Ilya Gutkovskiy, Feb 08 2021

			Triangle T(n,k) begins:
  1;
  0,  1;
  0,  0,  1;
  0,  1,  0,  1;
  0,  1,  2,  0,  1;
  0,  1,  2,  3,  0,  1;
  0,  1,  3,  3,  4,  0,  1;
  0,  1,  4,  6,  4,  5,  0,  1;
  0,  2,  5,  9, 10,  5,  6,  0,  1;
  0,  2,  8, 13, 16, 15,  6,  7,  0,  1;
  0,  2,  9, 21, 26, 25, 21,  7,  8,  0,  1;
  0,  2, 12, 27, 44, 45, 36, 28,  8,  9,  0,  1;
  ...

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

Columns k=0-10 give A000007, A000700, A338463, A341241, A341243, A341244, A341245, A341246, A341247, A341251, A341253.
Main diagonal and lower diagonals give A000012, A000004, A001477, A000217, A000290.
Row sums give A307058.
T(2n,n) gives A341265.
Cf. A000009, A047265, A060642, A286352, A308680.

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:
T:= proc(n, k) option remember;
      `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 09 2021

T[n_, k_] := SeriesCoefficient[(-1 + 2/QPochhammer[-1, -x])^k, {x, 0, n}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten

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

Sum_{k=0..n} (-1)^(n-k) * T(n,k) = A000009(n).

A047639 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^14 in powers of x.

Original entry on oeis.org

1, -14, 91, -364, 987, -1820, 1897, 754, -8008, 18928, -26845, 19460, 15015, -76272, 141065, -163072, 90727, 99386, -368277, 602616, -643734, 358190, 274547, -1101100, 1801086, -1982330, 1344525, 148316, -2163590, 4032756, -4938843, 4216576

Offset: 14

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 14..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Cf. A001482 - A001488, A001490, A047638, A047640 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(14) )); // G. C. Greubel, Sep 07 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*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=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 14):
seq(a(n), n=14..45);  # Alois P. Heinz, Feb 07 2021

nmax=45; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^14, {x,0,nmax}], x]//Drop[#, 14] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=14}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

my(x='x+O('x^40)); Vec((eta(-x)-1)^14) \\ Joerg Arndt, Sep 07 2023

from sage.modular.etaproducts import qexp_eta
m=75; k=14;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047639_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047639_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^14. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047640 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^15 in powers of x.

Original entry on oeis.org

1, -15, 105, -455, 1350, -2793, 3625, -765, -9840, 29120, -48657, 47370, 1680, -111060, 252555, -343526, 267540, 63210, -623510, 1216425, -1495173, 1093210, 166425, -2073645, 3963260, -4864839, 3872295, -618310, -4345470, 9477960, -12611991

Offset: 15

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 15..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Cf. A001482 - A001488, A001490, A047638, A047639, A047641 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(15) )); // G. C. Greubel, Sep 07 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*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=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 15):
seq(a(n), n=15..45);  # Alois P. Heinz, Feb 07 2021

nmax=45; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^15, {x, 0, nmax}], x]//Drop[#, 15] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=15}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

my(x='x+O('x^40)); Vec((eta(-x)-1)^15) \\ Joerg Arndt, Sep 07 2023

from sage.modular.etaproducts import qexp_eta
m=75; k=15;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047640_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047640_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^15. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047641 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^16 in powers of x.

Original entry on oeis.org

1, -16, 120, -560, 1804, -4128, 6312, -3920, -10530, 42208, -82752, 99584, -39460, -141200, 422568, -673936, 660941, -144720, -938840, 2301568, -3257188, 2916592, -628040, -3492160, 8217536, -11341568, 10408280, -3885040, -7668720, 21033408

Offset: 16

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 16..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Cf. A001482 - A001488, A001490, A047638 - A047640, A047642 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(16) )); // G. C. Greubel, Sep 07 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*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=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 16):
seq(a(n), n=16..45);  # Alois P. Heinz, Feb 07 2021

nmax=45; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] -1)^16, {x,0,nmax}], x]//Drop[#, 16] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=16}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

my(x='x+O('x^40)); Vec((eta(-x)-1)^16) \\ Joerg Arndt, Sep 07 2023

from sage.modular.etaproducts import qexp_eta
m=75; k=16;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047641_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047641_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^16. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047642 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^17 in powers of x.

Original entry on oeis.org

1, -17, 136, -680, 2363, -5916, 10319, -9656, -8534, 57426, -133076, 190383, -134810, -140148, 657611, -1240116, 1461337, -770917, -1171504, 4061946, -6678161, 7071269, -3376863, -4939180, 15963612, -25098443, 26265408, -14513461, -10810368, 43792034

Offset: 17

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 17..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Cf. A001482 - A001488, A001490, A047638 - A047641, A047643 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(17) )); // G. C. Greubel, Sep 07 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*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=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 17):
seq(a(n), n=17..46);  # Alois P. Heinz, Feb 07 2021

nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^17, {x, 0, nmax}], x]//Drop[#, 17] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=17}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

my(x='x+O('x^40)); Vec((eta(-x)-1)^17) \\ Joerg Arndt, Sep 07 2023

from sage.modular.etaproducts import qexp_eta
m=75; k=17;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047642_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047642_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^17. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047643 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^18 in powers of x.

Original entry on oeis.org

1, -18, 153, -816, 3042, -8262, 16098, -19278, -1377, 72556, -203184, 339030, -326961, -53244, 940050, -2147916, 2975391, -2293488, -911369, 6616332, -12906162, 15883884, -10936899, -4660974, 28758849, -52660134, 62518248, -44501988, -7465464, 84565242

Offset: 18

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 18..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Cf. A001482 - A001488, A001490, A047638 - A047642, A047644 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(18) )); // G. C. Greubel, Sep 07 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*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=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 18):
seq(a(n), n=18..47);  # Alois P. Heinz, Feb 07 2021

nmax=47; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] -1)^18, {x,0,nmax}], x]//Drop[#, 18] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=18}, Drop[CoefficientList[Series[(QPochhammer[-x]-1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

my(x='x+O('x^35)); Vec((eta(-x)-1)^18) \\ Joerg Arndt, Sep 07 2023

from sage.modular.etaproducts import qexp_eta
m=75; k=18;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047643_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047643_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^18. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047644 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^19 in powers of x.

Original entry on oeis.org

1, -19, 171, -969, 3857, -11286, 24206, -34542, 14706, 83011, -294880, 569753, -680694, 220286, 1198672, -3502612, 5661867, -5571579, 791350, 9721976, -23494393, 33415357, -29225230, 2352751, 47086598, -104517176, 140834118, -121255530

Offset: 19

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 19..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Cf. A001482 - A001488, A001490, A047638 - A047643, A047645 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(19) )); // G. C. Greubel, Sep 07 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*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=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 19):
seq(a(n), n=19..46);  # Alois P. Heinz, Feb 07 2021

nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^19, {x,0,nmax}], x]//Drop[#, 19] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=19}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

my(x='x+O('x^35)); Vec((eta(-x)-1)^19) \\ Joerg Arndt, Sep 07 2023

from sage.modular.etaproducts import qexp_eta
m=75; k=19;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047644_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047644_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^19. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047646 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^21 in powers of x.

Original entry on oeis.org

1, -21, 210, -1330, 5964, -19929, 50253, -91920, 97965, 51604, -526659, 1389297, -2280320, 2118690, 769065, -7613319, 17220042, -23999430, 18024405, 10748850, -63778953, 124134772, -152793270, 99072120, 71722224, -341062407, 610085721

Offset: 21

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 21..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.

Cf. A001482 - A001488, A001490, A047638 - A047645, A047647 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(21) )); // G. C. Greubel, Sep 06 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*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=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 21):
seq(a(n), n=21..47);  # Alois P. Heinz, Feb 07 2021

nmax=47; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] -1)^21, {x, 0, nmax}], x]//Drop[#, 21] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=21}, Drop[CoefficientList[Series[(QPochhammer[-x] - 1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 06 2023 *)

my(N=44, x='x+O('x^N)); Vec((eta(-x)-1)^21) \\ Joerg Arndt, Sep 06 2023

from sage.modular.etaproducts import qexp_eta
m=75; k=21;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047646_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047646_list(m); a[k:] # G. C. Greubel, Sep 06 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^21. - G. C. Greubel, Sep 06 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047647 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^22 in powers of x.

Original entry on oeis.org

1, -22, 231, -1540, 7293, -25872, 69971, -140822, 183711, -25102, -634480, 2027804, -3817814, 4439116, -919600, -9829270, 27660479, -44779042, 43632974, -1898820, -92518261, 219961214, -313463842, 267448104, 15757973, -547042056, 1173033400

Offset: 22

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 22..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.

Cf. A001482 - A001488, A001490, A047638 - A047646, A047648, A047649, A047654, A047655, A341243.

m:=75;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(22) )); // G. C. Greubel, Sep 05 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*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=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 22):
seq(a(n), n=22..48);  # Alois P. Heinz, Feb 07 2021

nmax=48; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^22, {x, 0, nmax}], x]//Drop[#, 22] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=22}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 05 2023 *)

my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^22) \\ Michel Marcus, Sep 05 2023

from sage.modular.etaproducts import qexp_eta
m=75; k=22;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047647_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047647_list(m); a[k:] # G. C. Greubel, Sep 05 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^22. - G. C. Greubel, Sep 05 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A341265 Coefficient of x^(2*n) in (-1 + Product_{k>=1} 1 / (1 + x^k))^n.

Original entry on oeis.org

1, 0, 2, 3, 10, 25, 71, 203, 562, 1650, 4667, 13673, 39427, 115440, 336639, 987628, 2898658, 8529257, 25134200, 74173606, 219207815, 648546314, 1921045953, 5695642513, 16902924883, 50203798050, 149229323544, 443895849894, 1321292939459, 3935377071154, 11728037768186

Offset: 0

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000700, A081362, A255526, A324595, A338463, A340987, A341241, A341243, A341244, A341245, A341246, A341247, A341251, A341253, A341263, A341279.

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=0, 1, `if`(k=1, g(n+1),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 07 2021

Table[SeriesCoefficient[(-1 + 1/QPochhammer[-x, x])^n, {x, 0, 2 n}], {n, 0, 30}]
A[n_, k_] := A[n, k] = If[n == 0, 1, -k Sum[A[n - j, k] Sum[Mod[d, 2] d, {d, Divisors[j]}], {j, 1, n}]/n]; T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}]; Table[T[2 n, n], {n, 0, 30}]

a(n) = A341279(2n,n).

a(n) ~ c * d^n / sqrt(n), where d = 3.03044218957412050685579849718626198523346... and c = 0.2319377657497495246637662111041144... - Vaclav Kotesovec, Feb 20 2021

cp's OEIS Frontend

A341279 Triangle read by rows: T(n,k) = coefficient of x^n in expansion of (-1 + Product_{j>=1} 1 / (1 + (-x)^j))^k, n >= 0, 0 <= k <= n.

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A047639 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^14 in powers of x.

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A047640 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^15 in powers of x.

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A047641 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^16 in powers of x.

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A047642 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^17 in powers of x.

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A047643 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^18 in powers of x.

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