A262308 -id:A262308 - cp's OEIS frontend

A008705 Coefficient of x^n in (Product_{m=1..n}(1-x^m))^n.

1, -1, -1, 5, -5, -6, 11, 41, -125, -85, 1054, -2069, -209, 8605, -15625, 3990, 14035, 36685, -130525, -254525, 1899830, -3603805, -134905, 13479425, -25499225, 23579969, -64447293, 237487433, -133867445, -1795846200, 6309965146, -6788705842, -11762712973

Offset: 0

T. Forbes (anthony.d.forbes(AT)googlemail.com)

sign

Degree of resulting polynomial is A002411(n). - Michel Marcus, Sep 05 2013
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k. - Peter Bala, Jan 31 2022
Conjectures: the supercongruences a(p) == -1 - p (mod p^2) and a(2*p) == p - 1 (mod p^2) hold for all primes p >= 3. - Peter Bala, Apr 18 2023

			(1-x)^1 = -x + 1, hence a(1) = -1.
(1-x^2)^2*(1-x)^2 = x^6 - 2*x^5 - x^4 + 4*x^3 - x^2 - 2*x + 1, hence a(2) = -1.

Seiichi Manyama, Table of n, a(n) for n = 0..2856 (terms 0..256 from N. J. A. Sloane)
Morris Newman, Further identities and congruences for the coefficients of modular forms, Canadian J. Math 10 (1958): 577-586. See Table 1, column p=5.
Morris Newman, Further identities and congruences for the coefficients of modular forms [annotated scanned copy], Canadian J. Math 10 (1958): 577-586. See Table 1, column p=5.

Bisections: A262308, A262309.

Main diagonal of A286354.

C5:=proc(r) local t1,n; t1:=mul((1-x^n)^r,n=1..r+2); series(t1,x,r+1); coeff(%,x,r); end;
[seq(C5(i),i=0..30)]; # N. J. A. Sloane, Oct 04 2015
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, -k*
      add(numtheory[sigma](j)*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 21 2018

With[{m = 40}, Table[SeriesCoefficient[Series[(Product[1-x^j, {j, n}])^n, {x, 0, m}], n], {n, 0, m}]] (* G. C. Greubel, Sep 09 2019 *)

a(n) = polcoeff(prod(m = 1, n, (1-x^m)^n), n); \\ Michel Marcus, Sep 05 2013

a(n) = [x^n] exp(-n*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, May 30 2018

More terms from Michel Marcus, Sep 05 2013

a(0)=1 prepended by N. J. A. Sloane, Oct 04 2015

A262309 Bisection of A008705.

Original entry on oeis.org

1, -1, -5, 11, -125, 1054, -209, -15625, 14035, -130525, 1899830, -134905, -25499225, -64447293, -133867445, 6309965146, -11762712973, -29646853725, -160561408097, -1495301415045, 24032249395590, -42497223256645, -176218120143125, 271151209742715, -6483574667232425

Offset: 0

N. J. A. Sloane, Oct 04 2015

sign

Cf. A008705, A262308.

A262538 Bisection of A262310.

Original entry on oeis.org

-1, -7, 9, -176, -673, 1143, 33201, 315783, 2210985, 1790369, 44496424, 1040502519, -1810005816, 7445790947, -134133273672, -627282760902, 7903957116411, -22364317249111, -200569974966918, 510174845699733, 15061484475966460, 42739249663260682, 758042190595492047

Offset: 0

N. J. A. Sloane, Oct 04 2015

sign

Morris Newman, Further identities and congruences for the coefficients of modular forms [annotated scanned copy], Canadian J. Math 10 (1958): 577-586. See Table 1, column p=7.

Cf. A262310, A262537.

A262541 Bisection of A262539.

Original entry on oeis.org

-3, -12, 1107, 33345, 151593, -17376120, -544200660, -2551628817, 298555990425, 9538198578972, 47571625958796, -5260288747038957, -171603777320158968, -932936371186135044, 93260709219956914584, 3120592373597958070140, 18795651984942284003265

Offset: 0

N. J. A. Sloane, Oct 04 2015

sign

Morris Newman, Further identities and congruences for the coefficients of modular forms [annotated scanned copy], Canadian J. Math 10 (1958): 577-586. See Table 1, column p=3.

Cf. A262539, A262540.

A262542 List of numbers arising in Theorem 5 of Morris Newman's "Further identities and congruences for the coefficients of modular forms".

Original entry on oeis.org

6, 10, 17, 18, 24, 27, 57, 68, 69, 74, 90, 95, 98, 103, 123, 127, 131, 163, 179, 197, 204, 210, 238, 239, 249, 250, 253, 256, 258, 259, 270, 274, 278, 282, 292, 326, 349, 359, 360, 364, 373, 374, 376, 378, 400, 407, 424, 425, 447, 448, 451, 454, 474, 480, 492, 493, 507, 558, 563, 569

Offset: 1

N. J. A. Sloane, Oct 04 2015

nonn

These are exactly the numbers n such that A010819(n) = 0 mod 13 and 24*n + 11 is squarefree. - Robin Visser, Jul 24 2023

Robin Visser, Table of n, a(n) for n = 1..5000
Morris Newman, Further identities and congruences for the coefficients of modular forms [annotated scanned copy], Canadian J. Math 10 (1958): 577-586. See Table 1, column p=5.

Cf. A010819.

for n in range(1, 1000):
    p11 = product([(1 - x^k)^11 for k in range(1, n+1)])
    p11n = int(p11.taylor(x, 0, n).coefficients()[n][0])
    if (p11n%13 == 0) and (24*n + 11).is_squarefree():
        print(n)  # Robin Visser, Jul 24 2023

More terms from Robin Visser, Jul 24 2023

cp's OEIS Frontend

A008705 Coefficient of x^n in (Product_{m=1..n}(1-x^m))^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A262309 Bisection of A008705.

Views

Author

Keywords

Crossrefs

A262538 Bisection of A262310.

Views

Author

Keywords

Links

Crossrefs

A262541 Bisection of A262539.

Views

Author

Keywords

Links

Crossrefs

A262542 List of numbers arising in Theorem 5 of Morris Newman's "Further identities and congruences for the coefficients of modular forms".

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Sage

Extensions