A077365 -id:A077365 - cp's OEIS frontend

A256125 Coefficients in asymptotic expansion of sequence A077365.

1, 1, 3, 12, 67, 457, 3734, 35741, 392875, 4886114, 67924417, 1044531625, 17609980356, 322993544751, 6402464186243, 136373115537072, 3105809328600351, 75300018326324541, 1936106590359322126, 52615058519875702993, 1506721174739412743551

Offset: 0

Vaclav Kotesovec, Mar 15 2015

nonn

			A077365(n) / n! ~ 1 + 1/n + 3/n^2 + 12/n^3 + 67/n^4 + 457/n^5 + 3734/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..50

Cf. A077365, A248871, A256124, A265954.

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

A107895 Euler transform of n!.

Original entry on oeis.org

1, 1, 3, 9, 36, 168, 961, 6403, 49302, 430190, 4199279, 45326013, 535867338, 6884000262, 95453970483, 1420538043009, 22579098396600, 381704267100888, 6837775526561031, 129377310771795789, 2578101967764973314, 53965231260126083854, 1183813954026245944519

Offset: 0

Thomas Wieder, May 26 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..170

Cf. A077365, A107894, A179327, A248871, A261047.

EulerTrans := proc(p) local b; b := proc(n) option remember; local d, j;
`if`(n=0,1, add(add(d*p(d),d=numtheory[divisors](j)) *b(n-j),j=1..n)/n) end end:
A107895 := EulerTrans(n->n!):  seq(A107895(n),n=0..20);
# After Alois P. Heinz, A000335.  [Peter Luschny, Jul 07 2011]

EulerTrans[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; A107895 = EulerTrans[Factorial]; Table[A107895[n], {n, 0, 22}] (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)

a(n) ~ n! * (1 + 1/n + 3/n^2 + 12/n^3 + 66/n^4 + 450/n^5 + 3679/n^6 + 35260/n^7 + 388511/n^8 + 4844584/n^9 + 67502450/n^10), for next coefficients see A248871. - Vaclav Kotesovec, Mar 14 2015

G.f.: Product_{n>=1} 1/(1-x^n)^(n!). - Vaclav Kotesovec, Aug 04 2015

A265950 Expansion of Product_{k>=1} (1 + k!*x^k).

Original entry on oeis.org

1, 1, 2, 8, 30, 156, 900, 6192, 47904, 422928, 4138848, 44864640, 531227520, 6836927040, 94891046400, 1413494219520, 22481104677120, 380261238681600, 6814832064422400, 128991143627965440, 2571187988206540800, 53834676521793638400, 1181214133296983654400

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A077365, A265954.

nmax=30; CoefficientList[Series[Product[(1+k!*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ n! * (1 + 1/n + 2/n^2 + 10/n^3 + 56/n^4 + 394/n^5 + 3332/n^6 + 32782/n^7 + 368072/n^8 + 4651666/n^9 + 65440748/n^10 + ...), for coefficients see A265954.

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*(j!)^k*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

A134134 Triangle of numbers obtained from the partition array A134133.

Original entry on oeis.org

1, 2, 1, 6, 2, 1, 24, 10, 2, 1, 120, 36, 10, 2, 1, 720, 204, 44, 10, 2, 1, 5040, 1104, 228, 44, 10, 2, 1, 40320, 7776, 1272, 244, 44, 10, 2, 1, 362880, 57600, 8760, 1320, 244, 44, 10, 2, 1, 3628800, 505440, 63936, 9096, 1352, 244, 44, 10, 2, 1

Offset: 1

Wolfdieter Lang, Oct 12 2007

In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

			[1];[2,1];[6,2,1];[24,10,2,1];[120,36,10,2,1];...
a(4,2)=10 from the sum over the numbers related to the partitions (1,3) and (2^2), namely
1!^1*3!^1 + 2!^2 = 6+4 = 10.

W. Lang, First 10 rows and more.

Row sums A077365. Alternating row sums A134135.

a(n,m)=sum(product(j!^e(n,m,k,j),j=1..n),k=1..p(n,m)) if n>=m>=1, else 0, with p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,k,j) is the exponent of j in the k-th m part partition of n.

A107107 For each partition of n, calculate (dM2/dM3) where dM2 = A036039(p) and dM3 = A036040(p); then sum over all partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 11, 37, 168, 926, 6181, 47651, 418546, 4106264, 44537519, 528408261, 6807428748, 94588717554, 1409927483625, 22437711255279, 379674820846534, 6806486383431340, 128862216628864163, 2569080120361323721, 53797824318887051264, 1180533584545138213222

Offset: 0

Alford Arnold, May 12 2005

Values for individual partitions (A107106) are factorials when all but one part of the partition has size one or two, but not usually in other cases.

			For n = 6, (120,144,90,40,90,120,15,40,45,15,1) / (1,6,15,10,15,60,15,20,45,15,1)
  equals (120,24,6,4,6,2,1,2,1,1,1) so A107107(6) = 168.

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

Cf. A000142, A036039, A000110, A036040, A107106, A102189.

Cf. A077365.

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
      `if`(i>n, 0, b(n-i, i)*(i-1)!)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 11 2016

nmax=20; CoefficientList[Series[Product[1/(1-(k-1)!*x^k),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 15 2015 *)

S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

For partition [], the contribution to the sum is product_i (c_i - 1)!^k_i.
G.f.: 1/Product_{m>0} (1-(m-1)!*x^m). - Vladeta Jovovic, Jul 10 2007
a(n) = S(n,1), where S(n,m) = sum(k=m..n/2, (k-1)!*S(n-k,k))+(n-1)!, S(n,n)=(n-1)!, S(0,m)=1, S(n,m)=0 for m>n. - Vladimir Kruchinin, Sep 07 2014
a(n) ~ (n-1)! * (1 + 1/n + 3/n^2 + 11/n^3 + 50/n^4 + 278/n^5 + 1861/n^6 + 14815/n^7 + 138477/n^8 + 1497775/n^9 + 18465330/n^10). - Vaclav Kotesovec, Mar 15 2015

Edited, corrected and extended by Franklin T. Adams-Watters, Nov 03 2005

More terms from Vladeta Jovovic, Jul 10 2007

A300520 Expansion of Product_{k>=1} 1 / (1 - Fibonacci(k)*x^k).

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 31, 57, 113, 212, 410, 757, 1464, 2684, 5083, 9380, 17569, 32120, 59977, 109193, 202046, 367951, 675541, 1224453, 2243795, 4052369, 7377243, 13314989, 24140198, 43406515, 78510429, 140800279, 253663615, 454352111, 815790813, 1457485309

Offset: 0

Vaclav Kotesovec, Mar 08 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4500

Cf. A000045, A023882, A075900, A077365, A179381, A318248.

nmax = 40; CoefficientList[Series[Product[1/(1-Fibonacci[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

log(a(n)) ~ log(phi)*n + 2*sqrt(polylog(2, 1/sqrt(5))*n) - 3*(log(n)/4), where polylog(2, 1/sqrt(5)) = 0.5107013915606224266804289751265205446721... and phi = A001622 = (1 + sqrt(5))/2 is the golden ratio.

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

A265954 Coefficients in asymptotic expansion of sequence A265950.

Original entry on oeis.org

1, 1, 2, 10, 56, 394, 3332, 32782, 368072, 4651666, 65440748, 1015205974, 17225810768, 317432262298, 6313880504564, 134828046043486, 3076458785723864, 74696205255843490, 1922729345267645180, 52297599798809376358, 1498690940537194229600

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

			A265950(n) / n! ~ 1 + 1/n + 2/n^2 + 10/n^3 + 56/n^4 + 394/n^5 + 3332/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..60

Cf. A077365, A256125, A265950.

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

A292279 Expansion of 1/Product_{k>=1} (1 + k!*x^k).

Original entry on oeis.org

1, -1, -1, -5, -15, -93, -551, -4129, -33607, -312929, -3179343, -35602881, -432201743, -5678740945, -80142780751, -1210609725905, -19481112885231, -332836223507793, -6016678424942063, -114746996449871761, -2302527084416470255, -48495552665272893329

Offset: 0

Seiichi Manyama, Sep 13 2017

sign

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A077365, A265950.

nmax = 25; CoefficientList[Series[Product[1/(1 + k!*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 13 2017 *)

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

Convolution inverse of A265950.
a(n) ~ -n! * (1 - 1/n - 1/n^2 - 6/n^3 - 31/n^4 - 219/n^5 - 1932/n^6 - 19945/n^7 - 234837/n^8 - 3104108/n^9 - 45495817/n^10). - Vaclav Kotesovec, Sep 14 2017
G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*(j!)^k*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

A292280 Expansion of Product_{k>=1} (1 - k!*x^k).

Original entry on oeis.org

1, -1, -2, -4, -18, -84, -564, -3984, -33504, -307728, -3156192, -35254080, -429350400, -5641133760, -79720588800, -1204747741440, -19400679325440, -331599765565440, -5996988417784320, -114408970262922240, -2296442484579686400, -48379417944213196800

Offset: 0

Seiichi Manyama, Sep 13 2017

sign

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A077365, A265950.

nmax = 25; CoefficientList[Series[Product[(1 - k!*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 13 2017 *)

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

Convolution inverse of A077365.
a(n) ~ -n! * (1 - 1/n - 2/n^2 - 6/n^3 - 32/n^4 - 222/n^5 - 1916/n^6 - 19650/n^7 - 231200/n^8 - 3058566/n^9 - 44883428/n^10). - Vaclav Kotesovec, Sep 14 2017
G.f.: exp(-Sum_{k>=1} Sum_{j>=1} (j!)^k*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

cp's OEIS Frontend

A256125 Coefficients in asymptotic expansion of sequence A077365.

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A107895 Euler transform of n!.

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A265950 Expansion of Product_{k>=1} (1 + k!*x^k).

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A134134 Triangle of numbers obtained from the partition array A134133.

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A107107 For each partition of n, calculate (dM2/dM3) where dM2 = A036039(p) and dM3 = A036040(p); then sum over all partitions of n.

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A300520 Expansion of Product_{k>=1} 1 / (1 - Fibonacci(k)*x^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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A265954 Coefficients in asymptotic expansion of sequence A265950.

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