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 1-10 of 10 results.

A379821 Array read by ascending antidiagonals: A(n, k) = (-1)^(n + k) * Sum_{j=0..k} (j!)^2 * Stirling1(n, j) * Stirling1(k, j).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 5, 2, 0, 0, 6, 14, 14, 6, 0, 0, 24, 50, 76, 50, 24, 0, 0, 120, 224, 360, 360, 224, 120, 0, 0, 720, 1216, 1908, 2392, 1908, 1216, 720, 0, 0, 5040, 7776, 11628, 15664, 15664, 11628, 7776, 5040, 0
Offset: 0

Views

Author

Peter Luschny, Jan 03 2025

Keywords

Examples

			Array begins:
  [0] 1,   0,    0,     0,      0,       0,        0,        0, ...
  [1] 0,   1,    1,     2,      6,      24,      120,      720, ...
  [2] 0,   1,    5,    14,     50,     224,     1216,     7776, ...
  [3] 0,   2,   14,    76,    360,    1908,    11628,    81072, ...
  [4] 0,   6,   50,   360,   2392,   15664,   110336,   856080, ...
  [5] 0,  24,  224,  1908,  15664,  126676,  1046780,  9169920, ...
  [6] 0, 120, 1216, 11628, 110336, 1046780, 10057204, 99846144, ...
.
Triangle T(n, k) = A(n - k, k) starts:
  [0] 1;
  [1] 0,   0;
  [2] 0,   1,    0;
  [3] 0,   1,    1,    0;
  [4] 0,   2,    5,    2,    0;
  [5] 0,   6,   14,   14,    6,    0;
  [6] 0,  24,   50,   76,   50,   24,    0;
  [7] 0, 120,  224,  360,  360,  224,  120,   0;
  [8] 0, 720, 1216, 1908, 2392, 1908, 1216, 720, 0;

Crossrefs

Main diagonal gives A382792.
The corresponding array with Stirling2 numbers is A371761.

Programs

  • Maple
    A := (n, k) -> local j; (-1)^(n + k)*add((j!)^2*Stirling1(n, j)*Stirling1(k, j), j = 0..k):
seq(lprint(seq(A(n, k), k = 0..7)), n = 0..8);
  • PARI
    a(n, k) = sum(j=0, min(n, k), j!^2*abs(stirling(n, j, 1)*stirling(k, j, 1))); \\ Seiichi Manyama, Apr 05 2025

Formula

E.g.f.: 1 / (1 - log(1-x) * log(1-y)). - Seiichi Manyama, Apr 05 2025

A382804 a(n) = Sum_{k=0..n} k! * (k+1)! * Stirling1(n,k)^2.

Original entry on oeis.org

1, 2, 14, 260, 9588, 581952, 52096512, 6423520896, 1041005447424, 214260350714496, 54547409318781312, 16820040059243046144, 6175245603727007034624, 2661063379044058584861696, 1329787781176741647226481664, 762665713456216694195942866944
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2025

Keywords

Crossrefs

Main diagonal of A382799.
Cf. A382792, A382806.
Cf. A382737.

Programs

  • PARI
    a(n) = sum(k=0, n, k!*(k+1)!*stirling(n, k, 1)^2);

Formula

a(n) == 0 (mod 2) for n > 0.
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1-x) * log(1-y))^2.
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1+x) * log(1+y))^2.
a(n) ~ sqrt(Pi) * n^(2*n + 3/2) / (exp(1) - 1)^(2*n+2). - Vaclav Kotesovec, Apr 05 2025

A382806 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling1(n,k)^2.

Original entry on oeis.org

1, 3, 27, 588, 24612, 1669128, 165049224, 22269896064, 3918921022656, 870149951146944, 237662482188210624, 78249086559726140160, 30547324837444471084800, 13946361918619108837939200, 7359961832428044552536217600, 4444946383758589481684168540160
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2025

Keywords

Crossrefs

Main diagonal of A382800.
Cf. A382792, A382804.
Cf. A382738.

Programs

  • PARI
    a(n) = sum(k=0, n, k!^2*binomial(k+2, 2)*stirling(n, k, 1)^2);

Formula

a(n) == 0 (mod 3) for n > 0.
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1-x) * log(1-y))^3.
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1+x) * log(1+y))^3.
a(n) ~ sqrt(Pi) * n^(2*n + 5/2) / (2 * (exp(1) - 1)^(2*n+3)). - Vaclav Kotesovec, Apr 05 2025

A192564 a(n) = Sum_{k=0..n} abs(Stirling1(n,k))*Stirling2(n,k)*(k!)^2.

Original entry on oeis.org

1, 1, 5, 74, 2186, 106524, 7703896, 773034912, 102673179360, 17429291711280, 3680338415133024, 945958227345434016, 290761516548473591232, 105309706114422166775040, 44384982810939832477305600, 21536846291826596564956445184
Offset: 0

Views

Author

Emanuele Munarini, Jul 04 2011

Keywords

Crossrefs

Cf. A047793, A048144, A382792.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n,k]]StirlingS2[n,k]k!^2,{k,0,n}],{n,0,100}]
nmax = 20; Table[SeriesCoefficient[1/(1 + (E^x - 1)*Log[1 - y]), {x, 0, n}, {y, 0, n}], {n, 0, nmax}] * Range[0, nmax]!^2 (* Vaclav Kotesovec, Apr 08 2025 *)
  • Maxima
    makelist(sum(abs(stirling1(n,k))*stirling2(n,k)*k!^2,k,0,n),n,0,24);

Formula

a(n) ~ c * LambertW(-1, -r*exp(-r))^n * n!^2 / (sqrt(n) * LambertW(-exp(-1/r)/r)^n), where r = 0.673313285145753168... is the root of the equation (1 + 1/(r*LambertW(-exp(-1/r)/r))) * (r + LambertW(-1, -r*exp(-r))) = 1 and c = 0.27034346270211507329954765593360596752557904498770241464597402478625037569... - Vaclav Kotesovec, Jul 05 2021
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 + (exp(x) - 1) * log(1 - y)). - Ilya Gutkovskiy, Apr 06 2025

A382794 a(n) = Sum_{k=0..n} Stirling1(n,k) * Stirling2(n,k) * (k!)^2.

Original entry on oeis.org

1, 1, 3, 2, -418, -14676, -234344, 18565056, 2659703616, 169046742960, -6539356064736, -4061128974843744, -672969012637199040, -19289566159655581440, 27323548725052131528960, 10157639436460221570630144, 1433264952547826545065237504, -520046813680980959472490690560
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 05 2025

Keywords

Crossrefs

Cf. A000670, A006252, A047792, A048144, A064618, A192554, A192564, A382792.

Programs

  • Mathematica
    Table[Sum[StirlingS1[n, k] StirlingS2[n, k] (k!)^2, {k, 0, n}], {n, 0, 17}]
Table[(n!)^2 SeriesCoefficient[1/(1 - (Exp[x] - 1) Log[1 + y]), {x, 0, n}, {y, 0, n}], {n, 0, 17}]

Formula

a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - (exp(x) - 1) * log(1 + y)).

A382807 a(n) = Sum_{k=0..n} (Stirling1(n,k) * k!)^3.

Original entry on oeis.org

1, 1, 7, 8, -22400, -3821176, 733375592, 1324952888832, 521577465629184, -1322687167356985344, -3493561791052460040192, 83811280007607865122816, 33603928402796871413168222208, 112696506862115060894313558528000, -389416384673353674591900391305326592
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 05 2025

Keywords

Crossrefs

Cf. A006252, A242280, A382792, A382808.

Programs

  • Mathematica
    Table[Sum[(StirlingS1[n, k] k!)^3, {k, 0, n}], {n, 0, 14}]
Table[(n!)^3 SeriesCoefficient[1/(1 - Log[1 + x] Log[1 + y] Log[1 + z]), {x, 0, n}, {y, 0, n}, {z, 0, n}], {n, 0, 14}]

Formula

a(n) = (n!)^3 * [(x*y*z)^n] 1 / (1 - log(1 + x) * log(1 + y) * log(1 + z)).

A382808 a(n) = Sum_{k=0..n} (|Stirling1(n,k)| * k!)^3.

Original entry on oeis.org

1, 1, 9, 440, 71344, 25826824, 17321581592, 19304140340736, 33142988156751360, 82906630912116006912, 289508760665893747703808, 1364207202603804952193826816, 8438589244471363680258331914240, 66972265137135031645961782287814656, 668922701586813036491303458870218731520
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 05 2025

Keywords

Comments

In general, for m>=1, Sum_{k=0..n} (abs(Stirling1(n,k)) * k!)^m ~ sqrt(2*Pi/m) * n^(m*n + 1/2) / (exp(1) - 1)^(m*n+1). - Vaclav Kotesovec, Apr 05 2025

Crossrefs

Cf. A007840, A242280, A382792, A382807.

Programs

  • Mathematica
    Table[Sum[(Abs[StirlingS1[n, k]] k!)^3, {k, 0, n}], {n, 0, 14}]
Table[(n!)^3 SeriesCoefficient[1/(1 + Log[1 - x] Log[1 - y] Log[1 - z]), {x, 0, n}, {y, 0, n}, {z, 0, n}], {n, 0, 14}]

Formula

a(n) = (n!)^3 * [(x*y*z)^n] 1 / (1 + log(1 - x) * log(1 - y) * log(1 - z)).
a(n) ~ sqrt(2*Pi/3) * n^(3*n + 1/2) / (exp(1) - 1)^(3*n+1). - Vaclav Kotesovec, Apr 05 2025

A382826 a(n) = Sum_{k=0..n} (k! * Stirling1(n+1,k+1))^2.

Original entry on oeis.org

1, 2, 17, 337, 12152, 696076, 58136500, 6673107316, 1008077743552, 193915431216576, 46281189562936704, 13420575661095930240, 4647502230640182602496, 1894412230202331489632256, 897850527136410029486517504, 489578762044356075253626875136
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2025

Keywords

Crossrefs

Main diagonal of A382823.
Cf. A048163, A382792.

Programs

  • PARI
    a(n) = sum(k=0, n, (k!*stirling(n+1, k+1, 1))^2);

Formula

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

A382853 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * (k! * Stirling1(n,k))^2.

Original entry on oeis.org

1, 1, 14, 588, 51064, 7542780, 1688795184, 532244030976, 224335607135616, 121793234373123840, 82750681453274478720, 68773648886955417943296, 68628724852793337500166144, 80970628401965472953705395200, 111490683570184861858636405923840, 177177650274516448010905794637332480
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2025

Keywords

Crossrefs

Cf. A382792, A382804, A382806.

Programs

  • Mathematica
    Table[Sum[Binomial[n+k-1,k] * k!^2 * StirlingS1[n,k]^2, {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Apr 07 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+k-1, k)*(k!*stirling(n, k, 1))^2);

Formula

a(n) == 0 (mod n) for n > 0.
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1-x) * log(1-y))^n.
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1+x) * log(1+y))^n.
a(n) ~ c * (r*(1+r) + sqrt(r*(1+r)))^(2*n) * n^(2*n) / (exp(2*n) * r^n), where r = 0.71197519729041875298209529969157574831688314013967... is the root of the equation (1+r)*(r + LambertW(-1, -r*exp(-r)))^2 = r and c = 0.61294561390083215776201123658816241786650851195222... - Vaclav Kotesovec, Apr 07 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 1-10 of 10 results.

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