A107895 -id:A107895 - cp's OEIS frontend

A248871 Coefficients in asymptotic expansion of sequence A107895.

1, 1, 3, 12, 66, 450, 3679, 35260, 388511, 4844584, 67502450, 1039929756, 17556193609, 322321551868, 6393505020803, 136245752898586, 3103879644045050, 75268872986970840, 1935571325829192247, 52605265683008056660, 1506530437404419817467

Offset: 0

Vaclav Kotesovec, Mar 14 2015

nonn

			A107895(n) / n! ~ 1 + 1/n + 3/n^2 + 12/n^3 + 66/n^4 + 450/n^5 + 3679/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..212

Cf. A107895, A256124.

a(k) ~ k! / (4 * (log(2))^(k+1)).

A179327 G.f.: Product_{n>=1} 1/(1-x^n)^((n-1)!).

Original entry on oeis.org

1, 1, 2, 4, 11, 37, 167, 925, 6164, 47630, 418227, 4105887, 44529413, 528398441, 6807143686, 94588353184, 1409913624333, 22437692156739, 379673925360239, 6806484898946045, 128862141334488784, 2569079946351669286, 53797816061915662161, 1180533553597621952193

Offset: 0

Paul D. Hanna, Jan 08 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 37*x^5 + 167*x^6 +...
A(x) = 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)^6*(1-x^5)^24*(1-x^6)^120*...).
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 27*x^4/4 + 121*x^5/5 + 729*x^6/6 + 5041*x^7/7 + 40347*x^8/8 +...+ A062363(n)*x^n/n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..448

Cf. A062363, A107895, A256126, A261047.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial((i-1)!+j-1, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 10 2021

nmax=20; CoefficientList[Series[Product[1/(1-x^k)^((k-1)!),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 14 2015 *)

{a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d!)*x^m/m)+x*O(x^n)),n)}

Euler transform of (n-1)!.
G.f.: A(x) = exp( Sum_{n>=1} A062363(n)*x^n/n ) where A062363(n) = Sum_{d|n} d!.
a(n) ~ (n-1)! * (1 + 1/n + 3/n^2 + 11/n^3 + 50/n^4 + 278/n^5 + 1860/n^6 + 14793/n^7 + 138166/n^8 + 1494034/n^9 + 18422609/n^10), for coefficients see A256126. - Vaclav Kotesovec, Mar 14 2015

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

Original entry on oeis.org

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).

A107894 Sum over the products of factorials of parts in all partitions of n where the sum runs over the number of different parts only.

Original entry on oeis.org

1, 1, 3, 9, 35, 167, 943, 6379, 48945, 429651, 4189865, 45307601, 535518109, 6883110373, 95435065935, 1420468921893, 22577620176887, 381695573051099, 6837601709298811, 129375694813679215, 2578070946813526485, 53964818587883937807, 1183805926540690127573

Offset: 0

Thomas Wieder, May 26 2005

nonn

			The partitions of 5 are 1+1+1+1+1, 1+1+1+2, 1+1+3, 1+2+2, 1+4, 2+3, 5, the corresponding products of factorials of parts are (when multiple parts are counted once only) 1!, 1!*2!, 1!*3!, 1!*2!, 1!*4!, 2!*3!, 5! and their sum is a(5) = 167.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A077365, A107895, A256124.

b:= proc(n, i) option remember;
      `if`(n=0 or i<2, 1, b(n, i-1) +i!*add(b(n-i*j, i-1), j=1..n/i))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 04 2012

Total[Times@@@(Union/@IntegerPartitions[#]!)]&/@Range[20]  (* Harvey P. Dale, Feb 26 2011 *)
b[n_, i_] := b[n, i] = If[n==0 || i<2, 1, b[n, i-1] + i!*Sum[b[n-i*j, i-1], {j, 1, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 29 2015, after Alois P. Heinz *)

a(n) ~ n! * (1 + 1/n + 3/n^2 + 12/n^3 + 65/n^4 + 443/n^5 + 3626/n^6 + 34811/n^7 + 384479/n^8 + 4806098/n^9 + 67109281/n^10), for coefficients see A256124. - Vaclav Kotesovec, Mar 15 2015

a(0) inserted and more terms from Alois P. Heinz, Apr 04 2012

A261047 Euler transform of (n+1)!.

Original entry on oeis.org

1, 2, 9, 40, 212, 1248, 8400, 63576, 540858, 5132564, 53952742, 623324184, 7855144818, 107224120980, 1575511525794, 24784246515256, 415435624535225, 7389692971336602, 138992875726543381, 2755750468146310688, 57433108983590606292

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Cf. A107895, A179327.

a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
     (d+1)!, d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 28 2017

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

a(n) ~ (n+1)! * (1 + 2/n + 7/n^2 + 33/n^3 + 219/n^4 + 1705/n^5 + 15707/n^6 + 166289/n^7 + 1993141/n^8 + 26727125/n^9 + 397081369/n^10).

a(n) ~ n! * n * (1 + 3/n + 9/n^2 + 40/n^3 + 252/n^4 + 1924/n^5 + 17412/n^6 + 181996/n^7 + 2159430/n^8 + 28720266/n^9 + 423808494/n^10).

A305867 Expansion of Product_{k>=1} 1/(1 - x^k)^(2*k-1)!!.

Original entry on oeis.org

1, 1, 4, 19, 130, 1120, 11960, 151595, 2230550, 37361755, 701873371, 14610774346, 333746628499, 8298025724194, 223049950124065, 6444634486214748, 199165237980655863, 6555102341516877027, 228905611339161301812, 8452656930719845696590, 329075775511339959533232, 13471099892869946627980017

Offset: 0

Ilya Gutkovskiy, Jun 12 2018

nonn

Euler transform of A001147.

Seiichi Manyama, Table of n, a(n) for n = 0..404
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Double Factorial
Index entries for sequences related to factorial numbers

Cf. A001147, A107895, A179327, A261047, A280088, A305868, A305869.

N:= 25:
S:=series(mul((1-x^k)^(-doublefactorial(2*k-1)),k=1..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Jun 12 2018

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

G.f.: Product_{k>=1} 1/(1 - x^k)^A001147(k).

A280088 Expansion of Product_{k>=1} 1/(1 - x^k)^(k!!).

Original entry on oeis.org

1, 1, 3, 6, 17, 38, 112, 280, 882, 2416, 8253, 24458, 91051, 289704, 1172288, 3980034, 17413820, 62706119, 294608079, 1118820630, 5603910081, 22328924231, 118432939871, 492897768426, 2752203529333, 11918139966134, 69709167028426, 313080284080648, 1910245872252972, 8873669214476627, 56283324138424814, 269790676411694902

Offset: 0

Ilya Gutkovskiy, Dec 25 2016

nonn

Euler transform of the double factorials (A006882).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Cf. A006882, A107895.

nmax = 31; CoefficientList[Series[Product[1/(1 - x^k)^(k!!), {k, 1, nmax}], {x, 0, nmax}], x]

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

a(n) ~ n!!. - Vaclav Kotesovec, Dec 25 2016

A321875 a(n) = Sum_{d|n} d*d!.

Original entry on oeis.org

1, 5, 19, 101, 601, 4343, 35281, 322661, 3265939, 36288605, 439084801, 5748023639, 80951270401, 1220496112085, 19615115520619, 334764638530661, 6046686277632001, 115242726706374263, 2311256907767808001, 48658040163569088701, 1072909785605898275299

Offset: 1

Ilya Gutkovskiy, Nov 20 2018

nonn

Inverse Möbius transform of A001563.

Seiichi Manyama, Table of n, a(n) for n = 1..448
N. J. A. Sloane, Transforms

Cf. A000142, A001563, A062363, A107895.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[k*Factorial(k)*x^k/(1 - x^k): k in [1..(m+2)]]) )); // G. C. Greubel, Nov 20 2018

Table[Sum[d d!, {d, Divisors[n]}], {n, 21}]
nmax = 21; Rest[CoefficientList[Series[Sum[k k! x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 21; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

a(n) = sumdiv(n, d, d*d!); \\ Michel Marcus, Nov 20 2018

s = sum(k*factorial(k)*x^k/(1-x^k) for k in (1..24));
(s/x).series(x, 21).coefficients(x, sparse=false) # Peter Luschny, Nov 21 2018

G.f.: Sum_{k>=1} k*k!*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k!)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} A001563(d).

A380497 Euler transform of primorial numbers.

Original entry on oeis.org

1, 2, 9, 46, 314, 3072, 37641, 603510, 11148030, 249327430, 7040987792, 216220333314, 7895699690498, 321315600822232, 13770543972819903, 644232544408157820, 33954066516677635554, 1994206929690480710244, 121461036181617491970561, 8111955386813996410196454, 574814471423312085719652432

Offset: 0

Ilya Gutkovskiy, Jan 25 2025

nonn

Eric Weisstein's World of Mathematics, Primorial.

Cf. A002110, A030009, A030012, A107895, A380498.

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      add(d*p(d), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 25 2025

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

G.f.: Product_{k>=1} 1 / (1 - x^k)^prime(k)#.

A053483 Euler transform of A029767.

Original entry on oeis.org

1, 4, 18, 114, 900, 8845, 103861, 1427122, 22486706, 399906140, 7922936720, 173013117604, 4127746294408, 106806183646594, 2978731438384738, 89065499057526433, 2842061902985159593, 96395720127638538076, 3462922846509648162418

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A107895.

Rest[CoefficientList[Series[Product[1/(1 - x^k)^((k-1)!*(2^k-1)), {k, 1, 20}], {x, 0, 20}], x]] (* Vaclav Kotesovec, Aug 07 2015 *)

cp's OEIS Frontend

A248871 Coefficients in asymptotic expansion of sequence A107895.

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A179327 G.f.: Product_{n>=1} 1/(1-x^n)^((n-1)!).

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

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A107894 Sum over the products of factorials of parts in all partitions of n where the sum runs over the number of different parts only.

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A261047 Euler transform of (n+1)!.

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A305867 Expansion of Product_{k>=1} 1/(1 - x^k)^(2*k-1)!!.

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Maple

Mathematica

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A280088 Expansion of Product_{k>=1} 1/(1 - x^k)^(k!!).

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A321875 a(n) = Sum_{d|n} d*d!.

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