A283263 -id:A283263 - cp's OEIS frontend

A283272 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is expansion of Product_{j>=1} (1-x^j)^(j^k) in power of x.

1, 1, -1, 1, -1, -1, 1, -1, -2, 0, 1, -1, -4, -1, 0, 1, -1, -8, -5, 0, 1, 1, -1, -16, -19, -1, 4, 0, 1, -1, -32, -65, -9, 21, 4, 1, 1, -1, -64, -211, -55, 127, 49, 7, 0, 1, -1, -128, -665, -285, 807, 500, 81, 3, 0, 1, -1, -256, -2059, -1351, 5179, 4809, 1038, 45

Offset: 0

Seiichi Manyama, Mar 04 2017

			Square array begins:
   1,  1,  1,   1,   1,    1, ...
  -1, -1, -1,  -1,  -1,   -1, ...
  -1, -2, -4,  -8, -16,  -32, ...
   0, -1, -5, -19, -65, -211, ...
   0,  0, -1,  -9, -55, -285, ...
   1,  4, 21, 127, 807, 5179, ...

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

Columns k=0-9 give A010815, A073592, A283263, A283264, A283271, A283336, A283337, A283338, A283339, A283340.
Row k=5 gives A281581.
Main diagonal gives A283333.
Cf. A144048.

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

A283271 Expansion of exp( Sum_{n>=1} -sigma_5(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -16, -65, -55, 807, 4809, 13135, 550, -169070, -862710, -2281174, -1221309, 20194565, 114391575, 346400092, 486546751, -1239516671, -11089537215, -41702958960, -93143227027, -45337210750, 674845109986, 3682196642725, 11405949184465, 20796945542222

Offset: 0

Seiichi Manyama, Mar 04 2017

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Let A(x) denote the g.f. and let m be an integer. Define a sequence by u(n) = [x^n] A(x)^(m*n). We conjecture that the supercongruence u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) holds for all positive integers n and r and all primes p >= 7. Cf. A380581. - Peter Bala, Jan 21 2025

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

Column k=4 of A283272.
Cf. A023873 (exp( Sum_{n>=1} sigma_5(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), this sequence (k=5).

G.f.: Product_{n>=1} (1 - x^n)^(n^4).

a(n) = -(1/n)*Sum_{k=1..n} sigma_5(k)*a(n-k).

A283264 Expansion of exp( Sum_{n>=1} -sigma_4(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -8, -19, -9, 127, 500, 1038, 448, -4967, -21463, -50043, -59084, 70418, 600080, 1837349, 3532062, 3179251, -6965009, -42260393, -119597290, -224546234, -223670132, 292245783, 2156083245, 6428174973, 13030612271, 16820582355, -133402359, -78307103593

Offset: 0

Seiichi Manyama, Mar 04 2017

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

Column k=3 of A283272.
Cf. A023872 (exp( Sum_{n>=1} sigma_4(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), this sequence (k=4), A283271 (k=5).

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -n^3)
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{n>=1} (1 - x^n)^(n^3).

a(n) = -(1/n)*Sum_{k=1..n} sigma_4(k)*a(n-k).

A380290 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} 1/(1 - x^k)^(k^2) is the g.f. of A023871.

Original entry on oeis.org

1, 1, 11, 73, 539, 3976, 30107, 229811, 1771803, 13749742, 107305836, 841211966, 6619647419, 52258136399, 413682035393, 3282569032273, 26101575743771, 207930807629248, 1659134361686186, 13258065574274885, 106084302933126364, 849845499077000534, 6815530442695480418, 54712839001004065090

Offset: 0

Peter Bala, Jan 19 2025

Given an integer sequence {f(n) : n >= 0} with f(0) = 1, there is a unique power series F(x) with rational coefficients, where F(0) = 1, such that f(n) = [x^n] F(x)^n. F(x) is given by F(x) = series_reversion(x/E(x)), where E(x) = exp(Sum_{n >= 1} f(n)*x^n/n). Furthermore, if the series E(x) has integer coefficients then the series F(x) also has integer coefficients and the sequence {f(n)} satisfies the Gauss congruences: f(n*p^r) == f(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r (by Stanley, Ch. 5, Ex. 5.2(a), p. 72 and the Lagrange inversion formula).
Thus the present sequence satisfies the Gauss congruences. In fact, stronger congruences appear to hold for the present sequence.
We conjecture that a(p) == 1 (mod p^3) for all primes p >= 7 (checked up to p = 61).
More generally, we conjecture that the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for all primes p >= 7 and positive integers n and r. Some examples are given below.

			Examples of supercongruences:
a(7) - a(1) = 229811 - 1 = 2*5*(7^3)*67 == 0 (mod 7^3)
a(3*7) - a(3) = 849845499077000534 - 73 = (7^3)*29243*84727410689 == 0 (mod 7^3)
a(19) - a(1) = 13258065574274885 - 1 = (2^2)*11*(19^3)*29*26723*56687 == 0 (mod 19^3)

R. P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Peter Bala, Notes on A380290 and A380291

Cf. A023871, A001158, A023873, A283263, A380291, A380581, A380582, A380583.

Cf. A008485, A255672, A386720.

with(numtheory):
G(x) := series(exp(add(sigma[3](k)*x^k/k, k = 1..23)),x,24):
seq(coeftayl(G(x)^n, x = 0, n), n = 0..23);

Table[SeriesCoefficient[Product[1/(1 - x^k)^(n*k^2), {k, 1, n}], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)
(* or *)
Table[SeriesCoefficient[Exp[n*Sum[DivisorSigma[3, k]*x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

a(n) = [x^n] exp(n*Sum_{k >= 1} sigma_3(k)*x^k/k).

a(n) ~ c * d^n / sqrt(n), where d = 8.20432131153340331179513077696629277558952852444670658917204305357709... and c = 0.2513708881073263860977360125648021910598660424705749139651716452651... - Vaclav Kotesovec, Jul 30 2025

A283336 Expansion of exp( Sum_{n>=1} -sigma_6(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -32, -211, -285, 5179, 44784, 162062, -125122, -5187417, -32587255, -95706881, 122837972, 3039216222, 17745876032, 52825817007, -24340390929, -1256623249600, -7805634068163, -26364952524572, -20649978457115, 368666542515083, 2777231006764690

Offset: 0

Seiichi Manyama, Mar 05 2017

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

Column k=5 of A283272.
Cf. A023874 (exp( Sum_{n>=1} sigma_6(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), A283271 (k=5), this sequence (k=6), A283337 (k=7), A283338 (k=8), A283339 (k=9), A283340 (k=10).

a[n_] := If[n<1, 1,-(1/n) * Sum[DivisorSigma[6, k] a[n - k], {k, n}]]; Table[a[n], {n, 0, 22}] (* Indranil Ghosh, Mar 16 2017 *)

a(n) = if(n<1, 1, -(1/n) * sum(k=1, n, sigma(k, 6) * a(n - k)));
for(n=0, 22, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 16 2017

G.f.: Product_{n>=1} (1 - x^n)^(n^5).

a(n) = -(1/n)*Sum_{k=1..n} sigma_6(k)*a(n-k).

A283337 Expansion of exp( Sum_{n>=1} -sigma_7(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -64, -665, -1351, 33111, 408149, 1959491, -4502590, -149420286, -1182474566, -3678670450, 22384197409, 377982157035, 2474860645111, 6161653683590, -48899064011245, -695916857379611, -4275491639488601, -10750056317745704, 69316545348329853

Offset: 0

Seiichi Manyama, Mar 05 2017

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

Column k=6 of A283272.
Cf. A023875 (exp( Sum_{n>=1} sigma_7(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), A283271 (k=5), A283336 (k=6), this sequence (k=7), A283338 (k=8), A283339 (k=9), A283340 (k=10).

G.f.: Product_{n>=1} (1 - x^n)^(n^6).

a(n) = -(1/n)*Sum_{k=1..n} sigma_7(k)*a(n-k).

A283338 Expansion of exp( Sum_{n>=1} -sigma_8(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -128, -2059, -6069, 210067, 3664420, 23366098, -116899962, -4133365357, -41809923367, -125160180169, 2447495850838, 42931762306584, 321967686614676, 281683012498569, -23874414003295851, -318729240693402530, -1992572289343189863

Offset: 0

Seiichi Manyama, Mar 05 2017

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

Column k=7 of A283272.
Cf. A023876 (exp( Sum_{n>=1} sigma_8(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), A283271 (k=5), A283336 (k=6), A283337 (k=7), this sequence (k=8), A283339 (k=9), A283340 (k=10).

G.f.: Product_{n>=1} (1 - x^n)^(n^7).

a(n) = -(1/n)*Sum_{k=1..n} sigma_8(k)*a(n-k).

A283339 Expansion of exp( Sum_{n>=1} -sigma_9(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -256, -6305, -26335, 1321887, 32565169, 276211695, -2659962750, -111341327890, -1454216029918, -3323783801026, 227018039015019, 4636828146319845, 39615489757794355, -132865771935151820, -9075288352543844755, -132703303201618610765

Offset: 0

Seiichi Manyama, Mar 05 2017

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

Column k=8 of A283272.
Cf. A023877 (exp( Sum_{n>=1} sigma_9(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), A283271 (k=5), A283336 (k=6), A283337 (k=7), A283338 (k=8), this sequence (k=9), A283340 (k=10).

G.f.: Product_{n>=1} (1 - x^n)^(n^8).

a(n) = -(1/n)*Sum_{k=1..n} sigma_9(k)*a(n-k).

A283340 Expansion of exp( Sum_{n>=1} -sigma_10(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -512, -19171, -111645, 8255899, 287477144, 3248973702, -56353404842, -2946880278857, -50078654012311, -24091665240825, 19437354184565824, 486126425619195338, 4607922953609319032, -63107867988829247005, -3101395214088243725145

Offset: 0

Seiichi Manyama, Mar 05 2017

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

Column k=9 of A283272.
Cf. A023878 (exp( Sum_{n>=1} sigma_10(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), A283271 (k=5), A283336 (k=6), A283337 (k=7), A283338 (k=8), A283339 (k=9), this sequence (k=10).

G.f.: Product_{n>=1} (1 - x^n)^(n^9).

a(n) = -(1/n)*Sum_{k=1..n} sigma_10(k)*a(n-k).

A307460 Expansion of Product_{k>=1} (1-x^k)^((-1)^k*k^2).

Original entry on oeis.org

1, 1, -3, 6, -4, -15, 54, -87, 63, 79, -405, 912, -1363, 1193, 510, -4900, 12512, -21582, 26512, -16540, -24585, 113682, -255045, 419931, -519210, 377176, 267957, -1703694, 4090424, -7179222, 9895981, -9897664, 3337614, 14790666, -49171217, 100903743

Offset: 0

Seiichi Manyama, Apr 09 2019

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k>=1} (1-x^k)^((-1)^k*k^b): A010054 (b=0), A281781 (b=1), this sequence (b=2).

Cf. A266964, A283263, A307462.

nmax = 40; CoefficientList[Series[Product[(1 - x^k)^((-1)^k*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 09 2019 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^((-1)^k*k^2)))

cp's OEIS Frontend

A283272 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is expansion of Product_{j>=1} (1-x^j)^(j^k) in power of x.

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A283271 Expansion of exp( Sum_{n>=1} -sigma_5(n)*x^n/n ) in powers of x.

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A283264 Expansion of exp( Sum_{n>=1} -sigma_4(n)*x^n/n ) in powers of x.

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SageMath

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A380290 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} 1/(1 - x^k)^(k^2) is the g.f. of A023871.

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Maple

Mathematica

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A283336 Expansion of exp( Sum_{n>=1} -sigma_6(n)*x^n/n ) in powers of x.

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Mathematica

PARI

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A283337 Expansion of exp( Sum_{n>=1} -sigma_7(n)*x^n/n ) in powers of x.

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A283338 Expansion of exp( Sum_{n>=1} -sigma_8(n)*x^n/n ) in powers of x.

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A283339 Expansion of exp( Sum_{n>=1} -sigma_9(n)*x^n/n ) in powers of x.

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A283340 Expansion of exp( Sum_{n>=1} -sigma_10(n)*x^n/n ) in powers of x.

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A307460 Expansion of Product_{k>=1} (1-x^k)^((-1)^k*k^2).

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