A261053 -id:A261053 - cp's OEIS frontend

A261052 Expansion of Product_{k>=1} (1+x^k)^(k!).

1, 1, 2, 8, 31, 157, 915, 6213, 48240, 423398, 4147775, 44882107, 531564195, 6837784087, 94909482330, 1413561537884, 22482554909451, 380269771734265, 6815003300096013, 128992737080703803, 2571218642722865352, 53835084737513866662, 1181222084520177393143

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Weigh transform of the factorial numbers. - Alois P. Heinz, Jun 11 2018

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000142, A107895, A168246, A261053, A026007, A027998, A248882, A102866, A256142.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i!, j)*b(n-i*j,i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 08 2015

nmax=25; CoefficientList[Series[Product[(1+x^k)^(k!),{k,1,nmax}],{x,0,nmax}],x]

seq(n)={Vec(exp(x*Ser(dirmul(vector(n, n, n!), -vector(n, n, (-1)^n/n)))))} \\ Andrew Howroyd, Jun 22 2018

a(n) ~ n! * (1 + 1/n + 2/n^2 + 10/n^3 + 57/n^4 + 401/n^5 + 3382/n^6 + 33183/n^7 + 371600/n^8 + 4685547/n^9 + 65792453/n^10).

A307497 Expansion of Product_{k>=1} (1+x^k)^((-1)^k*k^k).

Original entry on oeis.org

1, -1, 5, -32, 294, -3527, 51589, -894706, 17978610, -410803143, 10517824035, -298204099693, 9273022031794, -313755862498513, 11474175971184267, -450960476552715192, 18954545423649435646, -848383466771831169101, 40285210722052785437974

Offset: 0

Seiichi Manyama, Apr 10 2019

sign

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

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A083365, A261053, A266964, A284474, A307462.

nmax=20; CoefficientList[Series[Product[(1+x^k)^((-1)^k*k^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 12 2019 *)

N=20; x='x+O('x^N); Vec(prod(k=1, N, (1+x^k)^((-1)^k*k^k)))

a(n) ~ (-1)^n * n^n * (1 + exp(-1)/n + (exp(-1)/2 + 5*exp(-2))/n^2). - Vaclav Kotesovec, Apr 12 2019

A321387 Expansion of Product_{k>=1} (1 + x^k)^(k^(k-1)).

Original entry on oeis.org

1, 1, 2, 11, 74, 708, 8583, 127424, 2239965, 45514345, 1049365071, 27061132159, 771695223819, 24109698083919, 818914886275467, 30044684789498522, 1184048086192376822, 49883929845112421452, 2237287911899357657492, 106426388125032988691636, 5352033610656721914626572

Offset: 0

Ilya Gutkovskiy, Nov 08 2018

nonn

Weigh transform of A000169.

N. J. A. Sloane, Transforms

Cf. A000169, A023879, A261053, A283335, A321385, A321388.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1+x^k)^(k^(k-1)): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018

a:=series(mul((1+x^k)^(k^(k-1)),k=1..100),x=0,21): seq(coeff(a,x,n),n=0..20); # Paolo P. Lava, Apr 02 2019

nmax = 20; CoefficientList[Series[Product[(1 + x^k)^(k^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]

seq(n)={Vec(exp(sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^d ) * x^k/k) + O(x*x^n)))} \\ Andrew Howroyd, Nov 09 2018

m=30; x='x+O('x^m); Vec(prod(k=1,m,(1+x^k)^(k^(k-1)))) \\ G. C. Greubel, Nov 09 2018

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^d ) * x^k/k).

a(n) ~ n^(n-1) * (1 + exp(-1)/n + (3*exp(-1)/2 + 2*exp(-2))/n^2). - Vaclav Kotesovec, Nov 09 2018

A321388 Expansion of Product_{k>=1} (1 + x^k)^(k^(k-2)).

Original entry on oeis.org

1, 1, 1, 4, 19, 144, 1443, 18295, 280918, 5069651, 105147307, 2464296222, 64402891501, 1856989724951, 58560557062508, 2004999890781363, 74069439021212783, 2936703201134924845, 124383305232306494864, 5605027085651919547476, 267759074907470856179460, 13516676464234372267564939

Offset: 0

Ilya Gutkovskiy, Nov 08 2018

nonn

Weigh transform of A000272.

N. J. A. Sloane, Transforms

Cf. A000272, A261053, A262842, A321386, A321387.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 + x^k)^(k^(k-2)): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018

a:=series(mul((1+x^k)^(k^(k-2)),k=1..100),x=0,22): seq(coeff(a,x,n),n=0..21); # Paolo P. Lava, Apr 02 2019

nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(k^(k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]

m=30; x='x+O('x^m); Vec(prod(k=1,m,(1+x^k)^(k^(k-2)))) \\ G. C. Greubel, Nov 09 2018

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^(d-1) ) * x^k/k).

a(n) ~ n^(n-2) * (1 + exp(-1)/n + (5*exp(-1)/2 + exp(-2))/n^2). - Vaclav Kotesovec, Nov 09 2018

A304870 L.g.f.: log(Product_{k>=1} (1 + x^k)^(k^k)) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, 7, 82, 1015, 15626, 279862, 5764802, 134216695, 3486784483, 99999984382, 3138428376722, 106993205100070, 3937376385699290, 155568095552047430, 6568408355712906332, 295147905179218607095, 14063084452067724991010, 708235345355334189853093, 37589973457545958193355602

Offset: 1

Ilya Gutkovskiy, May 20 2018

nonn

			L.g.f.: L(x) = x + 7*x^2/2 + 82*x^3/3 + 1015*x^4/4 + 15626*x^5/5 + 279862*x^6/6 + 5764802*x^7/7 + 134216695*x^8/8 + 3486784483*x^9/9 + ...
exp(L(x)) = 1 + x + 4*x^2 + 31*x^3 + 289*x^4 + 3495*x^5 + 51268*x^6 + 891152*x^7 + 17926913*x^8 + 409907600*x^9 + ... + A261053(n)*x^n + ...

Index entries for sequences related to sums of divisors

Cf. A007778, A062796, A261053, A283498.

nmax = 19; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^k^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 19; Rest[CoefficientList[Series[Sum[k^(k + 1) x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Sum[(-1)^(n/d + 1) d^(d + 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 19}]

G.f.: Sum_{k>=1} k^(k+1)*x^k/(1 + x^k).
a(n) = Sum_{d|n} (-1)^(n/d+1)*d^(d+1).
a(p) = p^(p+1) + 1 where p is an odd prime.

A321389 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k^k).

Original entry on oeis.org

1, 2, 10, 72, 670, 7896, 113532, 1938948, 38463150, 869969602, 22098936536, 622728174288, 19271479902324, 649553475002720, 23680210649058960, 928276725059295192, 38931910620358040382, 1739307894106738293052, 82457731356894087128054, 4134332188240252347401752, 218571692793801915329820184

Offset: 0

Ilya Gutkovskiy, Nov 08 2018

nonn

Convolution of A023880 and A261053.

N. J. A. Sloane, Transforms

Cf. A023880, A156616, A206622, A206623, A206624, A261053.

a:=series(mul(((1+x^k)/(1-x^k))^(k^k),k=1..100),x=0,21): seq(coeff(a,x,n),n=0..20); # Paolo P. Lava, Apr 02 2019

nmax = 20; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[((-1)^(k/d + 1) + 1) d^(d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]

seq(n)={Vec(exp(sum(k=1, n, sumdiv(k,d, ((-1)^(k/d+1) + 1)*d^(d+1) ) * x^k/k) + O(x*x^n)))} \\ Andrew Howroyd, Nov 09 2018

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} ((-1)^(k/d+1) + 1)*d^(d+1) ) * x^k/k).

a(n) ~ 2 * n^n * (1 + 2*exp(-1)/n + (exp(-1) + 10*exp(-2))/n^2). - Vaclav Kotesovec, Nov 09 2018

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A261052 Expansion of Product_{k>=1} (1+x^k)^(k!).

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A307497 Expansion of Product_{k>=1} (1+x^k)^((-1)^k*k^k).

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A321387 Expansion of Product_{k>=1} (1 + x^k)^(k^(k-1)).

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A321388 Expansion of Product_{k>=1} (1 + x^k)^(k^(k-2)).

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A304870 L.g.f.: log(Product_{k>=1} (1 + x^k)^(k^k)) = Sum_{n>=1} a(n)*x^n/n.

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A321389 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k^k).

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