A283272 -id:A283272 - cp's OEIS frontend

A283333 Main diagonal of A283272.

1, -1, -4, -19, -55, 5179, 408149, 23366098, -2659962750, -2946880278857, -1715161696081878, 603927037021100215, 9904716216487281046207, 52286804207990141325901614, -71925062774291844591785748425, -17522340813140430159774329947096591

Offset: 0

Seiichi Manyama, Mar 04 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..80

Cf. A283272.

require 'prime'
def power(a, n)
  return 1 if n == 0
  k = power(a, n >> 1)
  k *= k
  return k if n & 1 == 0
  return k * a
end
def sigma(x, i)
  sum = 1
  pq = i.prime_division
  if x == 0
    pq.each{|a, n| sum *= n + 1}
  else
    pq.each{|a, n| sum *= (power(a, (n + 1) * x) - 1) / (power(a, x) - 1)}
  end
  sum
end
def A(k, m, n)
  ary = [1]
  s_ary = [0] + (1..n).map{|i| sigma(k, i * m)}
  (1..n).each{|i| ary << (1..i).inject(0){|s, j| s - ary[-j] * s_ary[j]} / i}
  ary
end
def A283333(n)
  (0..n).map{|i| A(i + 1, 1, i)[-1]}
end

a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(k^n). - Ilya Gutkovskiy, Mar 06 2018

A073592 Euler transform of negative integers.

Original entry on oeis.org

1, -1, -2, -1, 0, 4, 4, 7, 3, -2, -9, -17, -25, -24, -13, -1, 32, 61, 97, 111, 112, 74, 8, -108, -243, -392, -512, -569, -542, -358, -33, 473, 1078, 1788, 2395, 2865, 2955, 2569, 1496, -245, -2751, -5783, -9121, -12299, -14739, -15806, -14719, -10930, -3813, 6593, 20284, 36139, 53081, 68620, 80539

Offset: 0

Vladeta Jovovic, Aug 28 2002

sign

1/A(x) is g.f. for A000219.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
E. M. Wright, Coefficients of a reciprocal generating function, Quart. J. Math. 17 (1) (1966) 39-43, ADS Abstracts.
N. J. A. Sloane, Transforms

Column k=1 of A283272.

Cf. A000219, A001157, A001478, A026007, A156616, A255528.

a:= proc(n) option remember; `if`(n=0, 1, -add(
      numtheory[sigma][2](j)*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 12 2015

nmax=50; CoefficientList[Series[Exp[Sum[-x^k/(k*(1-x^k)^2),{k,1,nmax}]],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 02 2015 *)
a[n_]:= a[n] = -1/n*Sum[DivisorSigma[2,k]*a[n-k],{k,1,n}]; a[0]=1; Table[a[n],{n,0,100}] (* Vaclav Kotesovec, Mar 02 2015 *)

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -n)
print([b(n) for n in range(55)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{k>0} (1-x^k)^k.
a(n) = -1/n*Sum_{k=1..n} sigma[2](k)*a(n-k).
G.f.: exp( Sum_{n>=1} -sigma_2(n)*x^n/n ). - Seiichi Manyama, Mar 04 2017

A144048 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j^k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 5, 1, 1, 9, 14, 13, 7, 1, 1, 17, 36, 40, 24, 11, 1, 1, 33, 98, 136, 101, 48, 15, 1, 1, 65, 276, 490, 477, 266, 86, 22, 1, 1, 129, 794, 1828, 2411, 1703, 649, 160, 30, 1, 1, 257, 2316, 6970, 12729, 11940, 5746, 1593, 282, 42, 1, 1, 513

Offset: 0

Alois P. Heinz, Sep 08 2008

In general, column k > 0 is asymptotic to (Gamma(k+2)*Zeta(k+2))^((1-2*Zeta(-k)) /(2*k+4)) * exp((k+2)/(k+1) * (Gamma(k+2)*Zeta(k+2))^(1/(k+2)) * n^((k+1)/(k+2)) + Zeta'(-k)) / (sqrt(2*Pi*(k+2)) * n^((k+3-2*Zeta(-k))/(2*k+4))). - Vaclav Kotesovec, Mar 01 2015

			Square array begins:
  1,  1,   1,   1,    1,     1, ...
  1,  1,   1,   1,    1,     1, ...
  2,  3,   5,   9,   17,    33, ...
  3,  6,  14,  36,   98,   276, ...
  5, 13,  40, 136,  490,  1828, ...
  7, 24, 101, 477, 2411, 12729, ...

Alois P. Heinz, Antidiagonals = 0..99, flattened
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 21.
N. J. A. Sloane, Transforms

Columns k=0-9 give: A000041, A000219, A023871, A023872, A023873, A023874, A023875, A023876, A023877, A023878.
Rows give: 0-1: A000012, 2: A000051, A094373, 3: A001550, 4: A283456, 5: A283457.
Main diagonal gives A252782.
Cf. A283272.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0,1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->j^k)(n); seq(seq(A(n,d-n), n=0..d), d=0..13);

etr[p_] := Module[{ b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[Function[j, j^k]][n]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

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

A283263 Expansion of exp( Sum_{n>=1} -sigma_3(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -4, -5, -1, 21, 49, 81, 45, -121, -484, -997, -1344, -840, 1624, 6931, 15149, 23155, 23469, 2240, -57596, -168929, -322587, -461165, -450668, -64135, 985621, 2935044, 5718865, 8597971, 9683008, 5596899, -8414092, -37295629, -83336988, -141108721

Offset: 0

Seiichi Manyama, Mar 04 2017

sign

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

Column k=2 of A283272.
Cf. A023871 (exp( Sum_{n>=1} sigma_3(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), this sequence (k=3), A283264 (k=4), A283271 (k=5).

a[n_] := If[n<1, 1,-(1/n) * Sum[DivisorSigma[3, k] a[n - k], {k, n}]]; Table[a[n], {n, 0, 35}] (* Indranil Ghosh, Mar 16 2017 *)

a(n) = if(n<1, 1, -(1/n) * sum(k=1, n, sigma(k, 3) * a(n - k)));
for(n=0, 35, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 16 2017

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -n^2)
print([b(n) for n in range(36)]) # Peter Luschny, Nov 11 2020

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

a(n) = -(1/n)*Sum_{k=1..n} sigma_3(k)*a(n-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

sign

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

sign

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).

A284992 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)^(j^k) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 13, 8, 3, 1, 1, 16, 35, 31, 16, 4, 1, 1, 32, 97, 119, 83, 28, 5, 1, 1, 64, 275, 457, 433, 201, 49, 6, 1, 1, 128, 793, 1763, 2297, 1476, 487, 83, 8, 1, 1, 256, 2315, 6841, 12421, 11113, 4962, 1141, 142, 10, 1, 1

Offset: 0

Seiichi Manyama, Apr 07 2017

			Square array begins:
  1,  1,   1,    1,     1,      1,       1,        1, ...
  1,  1,   1,    1,     1,      1,       1,        1, ...
  1,  2,   4,    8,    16,     32,      64,      128, ...
  2,  5,  13,   35,    97,    275,     793,     2315, ...
  2,  8,  31,  119,   457,   1763,    6841,    26699, ...
  3, 16,  83,  433,  2297,  12421,   68393,   382573, ...
  4, 28, 201, 1476, 11113,  85808,  678101,  5466916, ...
  5, 49, 487, 4962, 52049, 561074, 6189117, 69540142, ...

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

Columns k=0-5 give A000009, A026007, A027998, A248882, A248883, A248884.
Rows (0+1),2-3 give: A000012, A000079, A007689.
Main diagonal gives A270917.
Cf. A283272, A284993.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i^k, j), j=0..n/i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Oct 16 2017

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n - i*j, i - 1, k]*Binomial[i^k, j], {j, 0, n/i}]]];
A[n_, k_] := b[n, n, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 10 2021, after Alois P. Heinz *)

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

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

sign

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

sign

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

sign

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).

cp's OEIS Frontend

A283333 Main diagonal of A283272.

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A073592 Euler transform of negative integers.

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A144048 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j^k).

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A283263 Expansion of exp( Sum_{n>=1} -sigma_3(n)*x^n/n ) in powers 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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A284992 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)^(j^k) in powers of x.

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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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