A292190 -id:A292190 - cp's OEIS frontend

A291698 a(n) = [x^n] Product_{k>=1} (1 + n*x^k).

1, 1, 2, 12, 20, 55, 294, 497, 1224, 2520, 14410, 21912, 54300, 104286, 220710, 1105215, 1697552, 3839382, 7356762, 14873580, 26275620, 132112596, 188666126, 423247104, 772560600, 1535398150, 2632049290, 4975242048, 21273166572, 30649985160, 64824339630, 116604788800, 223181224992

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

The number of partitions of n into distinct parts where each part can be colored in n different ways. For example, there are 4 partitions of 6 into distinct parts, namely 6, 5 + 1, 4 + 2 and 3 + 2 + 1; allowing for the colorings gives a(6) = 6 + 6*6 + 6*6 + 6*6*6 = 294. - Peter Bala, Aug 31 2017

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Robert Israel)

Main diagonal of A286957.

Cf. A022629, A124577, A292190, A292304.

seq(coeff(mul(1+n*x^k,k=1..n),x,n),n=0..50); # Robert Israel, Aug 30 2017

Table[SeriesCoefficient[Product[1 + n x^k, {k, 1, n}], {x, 0, n}], {n, 0, 32}]
Table[SeriesCoefficient[QPochhammer[-n, x]/(1 + n), {x, 0, n}], {n, 0, 32}]

a(n) = A286957(n,n).
a(n) == 0 (mod n); a(n) == n (mod n^2). - Peter Bala, Aug 31 2017
Conjecture: a(n) ~ exp(sqrt(2*(log(n)^2 + Pi^2/3)*n)) * (log(n)^2 + Pi^2/3)^(1/4) / (sqrt(Pi) * (2*n)^(5/4)). - Vaclav Kotesovec, Sep 15 2017

A265949 Expansion of Product_{k>=1} (1 + k^k*x^k).

Original entry on oeis.org

1, 1, 4, 31, 283, 3489, 50913, 890635, 17891170, 409850236, 10494427982, 297780829216, 9261266862273, 313453533534739, 11464487066049791, 450644378868285130, 18942868694407904729, 847930346323808122469, 40266107916200371331007, 2021842180288047801103956

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A023882, A292190, A292305, A292306.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+k^k*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

seq(coeff(series(mul((1+k^k*x^k),k=1..n),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018

nmax=20; CoefficientList[Series[Product[(1+k^k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

m=30; x='x+O('x^m); Vec(prod(k=1, m, (1+k^k*x^k))) \\ G. C. Greubel, Oct 31 2018

a(n) ~ n^n * (1 + exp(-1)/n + ((1/2)*exp(-1) + 4*exp(-2))/n^2).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d + 1)*d^(k+1) ) * x^k/k). - Ilya Gutkovskiy, Nov 08 2018

A292189 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j^k*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 13, 7, 3, 1, 1, 16, 35, 25, 15, 4, 1, 1, 32, 97, 91, 77, 25, 5, 1, 1, 64, 275, 337, 405, 161, 43, 6, 1, 1, 128, 793, 1267, 2177, 1069, 393, 64, 8, 1, 1, 256, 2315, 4825, 11925, 7313, 3799, 726, 120, 10

Offset: 0

Seiichi Manyama, Sep 11 2017

			Square array begins:
   1, 1,  1,  1,   1, ...
   1, 1,  1,  1,   1, ...
   1, 2,  4,  8,  16, ...
   2, 5, 13, 35,  97, ...
   2, 7, 25, 91, 337, ...

Alois P. Heinz, Rows n = 0..150, flattened

Columns k=0..5 give A000009, A022629, A092484, A265840, A265841, A265842.
Rows 0+1, 2, 3 give A000012, A000079, A007689.
Main diagonal gives A292190.
Cf. A292166.

b:= proc(n, i, k) option remember; (m->
      `if`(mn, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Sep 11 2017

m = 14;
col[k_] := col[k] = Product[1 + j^k*x^j, {j, 1, m}] + O[x]^(m+1) // CoefficientList[#, x]&;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)

A292305 a(n) = [x^n] Product_{k>=1} (1 + n^k*x^k).

Original entry on oeis.org

1, 1, 4, 54, 512, 9375, 186624, 4117715, 100663296, 3099363912, 100000000000, 3423740047332, 133741506723840, 5451751918660554, 244464150162276352, 11823135040283203125, 590295810358705651712, 31435129951680797038726, 1809934771463640728469504

Offset: 0

Vaclav Kotesovec, Sep 14 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A265949, A292190, A291698, A292306.

nmax = 20; Table[SeriesCoefficient[Product[(1+n^k*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
Flatten[{1, Table[n^n*PartitionsQ[n], {n, 1, 20}]}]

a(n) = n^n * A000009(n).

a(n) ~ exp(Pi*sqrt(n/3)) * n^(n-3/4) / (4*3^(1/4)).

A292194 Sum of n-th powers of products of terms in all partitions of n.

Original entry on oeis.org

1, 1, 5, 36, 610, 13225, 1173652, 92137513, 27960729094, 14612913824364, 11885159817456154, 23676862215173960082, 144210774157588042096815, 778807208565930895328294712, 15863318347221014170216633451982, 908978343753718115412387406378667615

Offset: 0

Seiichi Manyama, Sep 11 2017

nonn

			5 = 4 + 1 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1.
So a(5) = 5^5 + (4*1)^5 + (3*2)^5 + (3*1*1)^5 + (2*2*1)^5 + (2*1*1*1)^5 + (1*1*1*1*1)^5 = 13225.

Alois P. Heinz, Table of n, a(n) for n = 0..79

Main diagonal of A292193.

Cf. A292190.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      `if`(i>n, 0, i^k*b(n-i, i, k))+b(n, i-1, k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 11 2017

nmax = 20; Table[SeriesCoefficient[Product[1/(1 - k^n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 15 2017 *)

{a(n) = polcoeff(1/prod(k=1, n, 1-k^n*x^k+x*O(x^n)), n)}

a(n) = [x^n] Product_{k=1..n} 1/(1 - k^n*x^k).
From Vaclav Kotesovec, Sep 15 2017: (Start)
a(n) ~ 3^(n^2/3)               if mod(n,3)=0
a(n) ~ 3^(n*(n-4)/3)*2^(2*n+1) if mod(n,3)=1
a(n) ~ 3^(n*(n-2)/3)*2^n       if mod(n,3)=2
(End)

A292306 a(n) = [x^n] Product_{k>=1} (1 + n^n*x^k).

Original entry on oeis.org

1, 1, 4, 756, 65792, 19534375, 101564310279744, 558547898753326097, 9444733810164237336576, 174449211609498720646587480, 10000000004000000000400000000010000000000, 6626407607852766876000106671521201448502431912

Offset: 0

Vaclav Kotesovec, Sep 14 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..56

Cf. A265949, A292190, A292305.

nmax = 14; Table[SeriesCoefficient[Product[(1+n^n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]

Conjecture: log(a(n)) ~ (sqrt(2)*n^(3/2) - n/2)*log(n). - Vaclav Kotesovec, Aug 22 2018

A292072 Main diagonal of A292068.

Original entry on oeis.org

1, -1, -3, -20, 66, 4439, 454420, 4873175, -3803048954, -7320203267692, -1403057989033446, 6669491545211096686, 78492109668913945526447, 69591502229308312804788424, -6243846072108996200105800383026, -604234376454072219680822138902122079

Offset: 0

Seiichi Manyama, Sep 12 2017

sign

Alois P. Heinz, Table of n, a(n) for n = 0..79

Cf. A292190, A292194.

b:= proc(n, i, k) option remember; (m->
      `if`(mn, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
    end:
g:= proc(n,k) option remember; `if`(n=0, 1,
      -add(b(n-i$2, k)*g(i, k), i=0..n-1))
    end:
a:= n-> g(n$2):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 12 2017

b[n_, i_, k_] := b[n, i, k] = Function[m, If[m < n, 0, If[n == m, i!^k, b[n, i - 1, k] + If[i > n, 0, i^k*b[n - i, i - 1, k]]]]][i*(i + 1)/2];
g[n_, k_] := g[n, k] = If[n == 0, 1, -Sum[b[n-i, n-i, k]*g[i, k], {i, 0, n-1}]];
a[n_] := g[n, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 03 2018, after Alois P. Heinz *)

{a(n) = polcoeff(1/prod(k=1, n, 1+k^n*x^k+x*O(x^n)), n)}

from sympy.core.cache import cacheit
from sympy import factorial as f
@cacheit
def b(n, i, k):
    m=i*(i + 1)/2
    return 0 if mn else i**k*b(n - i, i - 1, k))
@cacheit
def g(n, k): return 1 if n==0 else -sum([b(n - i, n - i, k)*g(i, k) for i in range(n)])
def a(n): return g(n, n)
print([a(n) for n in range(16)]) # Indranil Ghosh, Sep 14 2017, after Maple program

a(n) = [x^n] Product_{k=1..n} 1/(1 + k^n*x^k).

cp's OEIS Frontend

A291698 a(n) = [x^n] Product_{k>=1} (1 + n*x^k).

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A265949 Expansion of Product_{k>=1} (1 + k^k*x^k).

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A292189 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j^k*x^j).

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A292305 a(n) = [x^n] Product_{k>=1} (1 + n^k*x^k).

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A292194 Sum of n-th powers of products of terms in all partitions of n.

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A292306 a(n) = [x^n] Product_{k>=1} (1 + n^n*x^k).

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A292072 Main diagonal of A292068.

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