cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A064618 Stirling transform of (n!)^2.

Original entry on oeis.org

1, 1, 5, 49, 821, 21121, 775205, 38516689, 2490976661, 203419086241, 20474978755205, 2490729330118129, 360263844701062901, 61114158974786823361, 12017074366801186956005, 2711409826920884006692369, 695820350706240448128979541, 201526362605605903609254528481
Offset: 0

Views

Author

Karol A. Penson, Sep 26 2001

Keywords

Comments

From Thomas Wieder, Oct 21 2004: (Start)
"Also the number of hierarchies with labeled elements and labeled levels where the levels are permuted. Let l_x denote level x, e.g. l_2 is level 2. Let 1 denote an element and 2 a second element and so on. Then l_1:123 means elements 1,2 and 3 are on level 1.
"Let | indicate separation between levels. Then l_1:1|l_2:346|l_3:5 denotes a hierarchy of n=6 unlabeled elements with element 1 on level 1, elements 3,4 and 6 on level 2 and element 5 on level 3.
"E.g. for n=3 one has a(3) = 49 possible hierarchies:
"l_1:123,
"l_1:12|l_2:3, l_1:13|l_2:2, l_1:23|l_2:1,
"l_2:12|l_1:3, l_2:13|l_1:2, l_2:23|l_1:1,
"l_1:1|l_2:23, l_1:2|l_2:13, l_1:3|l_2:12,
"l_2:1|l_1:23, l_2:2|l_1:13, l_2:3|l_1:12,
"l_1:1|l_2:2|l_3:3 and further five permutations of the elements with levels fixed,
"l_3:1|l_1:2|l_2:3 and further five permutations of the elements with levels fixed,. etc., up to
"l_3:1|l_2:2|l_1:3 and further five permutations of the elements with levels fixed. this gives 1 + 6 +6 + 6*6 = 49 = a(3) possible hierarchies.
"See A001339 for the number of hierarchies with unlabeled elements and labeled levels."
(End)
Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 10 the sequence becomes (1, 1, 5, 9, 1, 1, 5, 9, ...), with an apparent period 1, 1, 5, 9 of length 4 = phi(10) beginning at a(0). - Peter Bala, Jan 15 2018

Links

Crossrefs

Cf. A001044.
Cf. A001339.
Cf. A242282, A242283, A019774, A320502.

Programs

  • Maple
    a:= n-> add(Stirling2(n, k)*(k!^2), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 21 2012
  • Mathematica
    Table[Sum[(k!)^2*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 10 2014 *)
  • PARI
    /* By Vladeta Jovovic's formula: */
{a(n) = my(X=x+x*O(x^n)); n!*polcoeff( sum(m=0,n, m!*(exp(X)-1)^m), n)} /* Paul D. Hanna, Feb 15 2012 */

Formula

a(n) = Sum_{k=0..n} Stirling2(n, k)*(k!)^2.
E.g.f: hypergeom([1, 1], [], exp(x)-1). - Vladeta Jovovic, Sep 14 2003
O.g.f.: Sum_{n>=0} n!^2 * Product_{k=1..n} x/(1 - k*x). - Paul D. Hanna, Nov 25 2012
a(n) ~ exp(1/2) * (n!)^2. - Vaclav Kotesovec, May 10 2014

A192554 a(n) = Sum_{k=0..n} abs(Stirling1(n,k))*(-1)^(n-k)*k!^2.

Original entry on oeis.org

1, 1, 3, 26, 398, 9724, 344236, 16663968, 1056631824, 84962783664, 8446120969104, 1016998946575776, 145848462866589600, 24562489788256472064, 4799789988678066147840, 1077128972416478325901824, 275111625956753684599202304
Offset: 0

Views

Author

Emanuele Munarini, Jul 04 2011

Keywords

Links

Crossrefs

Cf. A006252, A320502, A351280.
Cf. A064618.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n,k]](-1)^(n-k)k!^2,{k,0,n}],{n,0,100}]
Table[Sum[StirlingS1[n,k] * k!^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 05 2021 *)
  • Maxima
    makelist(sum(abs(stirling1(n,k))*(-1)^(n-k)*k!^2,k,0,n),n,0,24);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, k!*log(1+x)^k))) \\ Seiichi Manyama, Apr 22 2022

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * k!^2. - Vaclav Kotesovec, Jul 05 2021
a(n) ~ exp(-1/2) * n!^2. - Vaclav Kotesovec, Jul 05 2021
E.g.f.: Sum_{k>=0} k! * log(1+x)^k. - Seiichi Manyama, Apr 22 2022

A385751 a(n) = Sum_{k=0..n} |Stirling1(n,k)| * (n!/k!)^2.

Original entry on oeis.org

1, 1, 5, 100, 5137, 539851, 101035441, 30669875230, 14117057058945, 9364637252286181, 8603755430968248301, 10603853731438585516856, 17077610933602804111318705, 35160631271792580418277658415, 90839446923946068488317221868825, 289828370988497912073923950177143826, 1126236403418687405801564385561640043521
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 08 2025

Keywords

Crossrefs

Cf. A119390, A320502, A385750, A385752.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n, k]] (n!/k!)^2, {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[Sum[(-Log[1 - x])^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3

Formula

Sum_{n>=0} a(n) * x^n / n!^3 = Sum_{k>=0} (-log(1 - x))^k / k!^3.
a(n) ~ (sqrt(2*Pi/3) * exp(3*log(n)^(1/3) - 3*n) * n^(3*n + 1/2) / log(n)) * (1 - 2/(9*log(n)^(1/3)) + (gamma - 4/81)/log(n)^(2/3) - (40/2187 + 11*gamma/9)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 09 2025

A382805 a(n) = Sum_{k=0..n} (-1)^(n-k) * (Stirling1(n,k) * k!)^2.

Original entry on oeis.org

1, 1, 3, 4, -272, -8524, -96596, 9634752, 983055168, 36429411456, -4303305703296, -1051644384152064, -89651253435644160, 10632887072757561600, 5599203549778990667520, 914684633796830925275136, -89559567563652079025946624, -104514775371103880549281775616
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 05 2025

Keywords

Crossrefs

Cf. A006252, A007840, A047796, A048144, A192554, A320502, A382792, A382793.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - k) (StirlingS1[n, k] k!)^2, {k, 0, n}], {n, 0, 17}]
Table[(n!)^2 SeriesCoefficient[1/(1 + Log[1 + x] Log[1 - y]), {x, 0, n}, {y, 0, n}], {n, 0, 17}]

Formula

a(n) = (n!)^2 * [(x*y)^n] 1 / (1 + log(1 + x) * log(1 - y)).
Showing 1-4 of 4 results.

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