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.

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A376765 a(n) = (1/2)*Sum_{k=0..n} n^binomial(n,k).

Original entry on oeis.org

0, 1, 4, 30, 2308, 9768755, 1828549405062726, 378818692266223327546801733500, 822752278660977165496641302425735395827886114383655917217382408, 1716153733051169540307898602341497569311487178262131715007420471535292324238528850823190109780802970900137357654221203141
Offset: 0

Views

Author

N. J. A. Sloane, Nov 02 2024

Keywords

Comments

For n>0, this is one-half of (one possible definition of) the number of partial maps from an n-set to itself.

Crossrefs

Cf. A000312, A007840, A376766.

Programs

  • Mathematica
    Table[Sum[n^Binomial[n,k],{k,0,n}]/2,{n,0,9}] (* James C. McMahon, Nov 03 2024 *)
  • Python
    from math import comb
def A376765(n): return sum(n**comb(n,k) for k in range(n+1))>>1 # Chai Wah Wu, Nov 03 2024

Extensions

a(9) from James C. McMahon, Nov 03 2024

A376766 a(n) = 1 + Sum_{k=1..n, j=1..k} binomial(n,k)*binomial(n,j)*|Stirling_1(k,j)|*j!.

Original entry on oeis.org

1, 2, 9, 67, 709, 9766, 165751, 3342081, 78023905, 2069303986, 61440372701, 2018742611535, 72713594116285, 2848845086153782, 120610707912196867, 5486918880456879061, 266925386719765703169, 13827085272988988990146, 759855686741314297312177, 44152359275709028329389627
Offset: 0

Views

Author

Peter J. Cameron, Nov 03 2024

Keywords

Comments

If Stirling_1 in the definition is changed to Stirling_2, we get A000169.

Links

Crossrefs

Cf. A000169, A000312, A002720, A007840, A376765.

Programs

  • Maple
    A376766 := proc(n) local k,j;
1 + sum(sum(binomial(n,k)*binomial(n,j)*abs(stirling1(k,j))*j!,j=1..k),k=1..n);
end; # N. J. A. Sloane, Nov 03 2024
  • Mathematica
    A376766[n_] := 1 + Sum[Binomial[n, k]*Binomial[n, j]*Abs[StirlingS1[k, j]]*j!, {k, n}, {j, k}];
Array[A376766, 25, 0] (* Paolo Xausa, Nov 04 2024 *)

Formula

a(n) ~ c * d^n * n^n / exp(n), where d = A226572 = -LambertW(-1, -exp(-2)) and c = 1.350274261169912007066341887216772613236351893372220769387... - Vaclav Kotesovec, Nov 09 2024

Extensions

Definition corrected by N. J. A. Sloane, Nov 09 2024

A377426 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^4)).

Original entry on oeis.org

1, 1, 11, 254, 9096, 443874, 27487034, 2065181880, 182545878152, 18562391987880, 2134764133508832, 273978733525211472, 38820518588599921200, 6019219063397716575840, 1013766602891962529642832, 184300120562198063868474624, 35971439241165448281366023424
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2024

Keywords

Crossrefs

Cf. A007840, A052802, A367138, A367139.
Cf. A377427, A377429.

Programs

  • PARI
    a(n) = sum(k=0, n, (4*n+k)!*abs(stirling(n, k, 1)))/(4*n+1)!;

Formula

a(n) = (1/(4*n+1)!) * Sum_{k=0..n} (4*n+k)! * |Stirling1(n,k)|.

A377494 E.g.f. satisfies A(x) = 1/(1 + A(x)^2 * log(1 - x*A(x)^2)).

Original entry on oeis.org

1, 1, 11, 248, 8632, 408794, 24550512, 1788220664, 153204336480, 15097630639464, 1682516996213376, 209233809698022240, 28725012833286981456, 4315256340778010888688, 704140465438516958644512, 124020015235118786512297728, 23450965881108082875087150336, 4738390708952218941582313234176
Offset: 0

Views

Author

Seiichi Manyama, Oct 29 2024

Keywords

Crossrefs

Cf. A007840, A367159, A377497.
Cf. A377492.

Programs

  • PARI
    a(n) = sum(k=0, n, (2*n+3*k)!/(2*n+2*k+1)!*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (2*n+3*k)!/(2*n+2*k+1)! * |Stirling1(n,k)|.

A377497 E.g.f. satisfies A(x) = 1/(1 + A(x)^3 * log(1 - x*A(x)^3)).

Original entry on oeis.org

1, 1, 15, 473, 23194, 1552084, 131908394, 13608546720, 1652258848656, 230829590868312, 36477894965606568, 6433858542834018240, 1252941162992516179776, 267027040073238416997024, 61819211233387513530840048, 15449035083850090935613775808, 4145148327496835979697002921216
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A007840, A367159, A377494.

Programs

  • PARI
    a(n) = sum(k=0, n, (3*n+4*k)!/(3*n+3*k+1)!*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (3*n+4*k)!/(3*n+3*k+1)! * |Stirling1(n,k)|.

A377692 E.g.f. satisfies A(x) = (1 - log(1 - x) * A(x))^2.

Original entry on oeis.org

1, 2, 12, 118, 1634, 29408, 654040, 17362056, 536410200, 18922946928, 750902659200, 33118793900784, 1607673329621712, 85192554602094912, 4894219487974911552, 303021216528999244416, 20116223556200658052992, 1425479651299747192856832, 107400336067263661850548224
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2024

Keywords

Crossrefs

Cf. A007840, A377693.
Cf. A052803, A377445.

Programs

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

Formula

E.g.f.: 4/(1 + sqrt(1 + 4*log(1-x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A052803.
a(n) = 2 * Sum_{k=0..n} (2*k+1)!/(k+2)! * |Stirling1(n,k)|.
a(n) ~ 2^(7/2) * n^(n-1) / ((exp(1/4) - 1)^(n - 1/2) * exp(3*n/4)). - Vaclav Kotesovec, Aug 27 2025

A377693 E.g.f. satisfies A(x) = (1 - log(1 - x) * A(x))^3.

Original entry on oeis.org

1, 3, 27, 408, 8814, 249702, 8789946, 370639896, 18233312640, 1025931258264, 65016004033944, 4583861319427200, 355955157532869552, 30192068409536580336, 2777615578746538933392, 275502517287785484635520, 29308962522270448504338048, 3329136621436554585165282048
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2024

Keywords

Crossrefs

Cf. A007840, A377692.
Cf. A367158, A377446.

Programs

  • PARI
    a(n) = 3*sum(k=0, n, (3*k+2)!/(2*k+3)!*abs(stirling(n, k, 1)));

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A367158.
a(n) = 3 * Sum_{k=0..n} (3*k+2)!/(2*k+3)! * |Stirling1(n,k)|.

A306037 Expansion of e.g.f. 1/(1 + log(1 - log(1 + x))).

Original entry on oeis.org

1, 1, 2, 7, 31, 178, 1200, 9588, 86592, 887086, 10035164, 125472246, 1705102394, 25175822644, 399387494956, 6801042408728, 123348694663480, 2379855020533664, 48569042602254128, 1047134236970183664, 23748242269316806752, 565834452464428045872, 14117321495269290091440
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 17 2018

Keywords

Examples

			1/(1 + log(1 - log(1 + x))) = 1 + x + 2*x^2/2! + 7*x^3/3! + 31*x^4/4! + 178*x^5/5! + 1200*x^6/6! + ...

Links

Crossrefs

Cf. A007840, A089064, A305323, A305988.

Programs

  • Maple
    a:=series(1/(1+log(1-log(1+x))),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 + Log[1 - Log[1 + x]]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[StirlingS1[n, k] Abs[StirlingS1[k, j]] j!, {j, 0, k}], {k, 0, n}], {n, 0, 22}]
a[0] = 1; a[n_] := a[n] = Sum[Sum[(j - 1)! StirlingS1[k, j], {j, 1, k}] a[n - k]/k!, {k, 1, n}]; Table[n! a[n], {n, 0, 22}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A007840(k).
a(n) ~ n! * exp(-exp(-1)) / (exp(1 - exp(-1)) - 1)^(n+1). - Vaclav Kotesovec, Jul 01 2018

A331798 E.g.f.: -log(1 - x) / ((1 - x) * (1 + log(1 - x))).

Original entry on oeis.org

0, 1, 5, 29, 204, 1714, 16862, 190826, 2447512, 35136696, 558727872, 9754239648, 185546362416, 3820734689472, 84687887312688, 2010622152615504, 50908186083448320, 1369376758488222336, 38998680958184088960, 1172297572938013827456, 37092793335394301708544
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 26 2020

Keywords

Crossrefs

Cf. A000254, A001008, A002805, A003713, A007526, A007840, A215916, A331797.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[-Log[1 - x]/((1 - x) (1 + Log[1 - x])), {x, 0, nmax}], x] Range[0, nmax]!
A007526[n_] := n! Sum[1/k!, {k, 0, n - 1}]; a[n_] := Sum[Abs[StirlingS1[n, k]] A007526[k], {k, 0, n}]; Table[a[n], {n, 0, 20}]
A007840[n_] := Sum[Abs[StirlingS1[n, k]] k!, {k, 0, n}]; a[n_] := Sum[Binomial[n, k] k! HarmonicNumber[k] A007840[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

a(n) = Sum_{k=0..n} = |Stirling1(n,k)| * A007526(k).
a(n) = Sum_{k=1..n} binomial(n,k) * k! * H(k) * A007840(n-k), where H(k) is the k-th harmonic number.
a(n) ~ n! / (1 - exp(-1))^(n+1). - Vaclav Kotesovec, Jan 26 2020

A336243 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * (n-k-1)! * a(k).

Original entry on oeis.org

1, 1, 5, 56, 1114, 34624, 1549648, 94402356, 7511324448, 756406501200, 94039208461584, 14146468841290752, 2532586289913605088, 532113978869395649856, 129662518122880634567232, 36270261084908437106586624, 11543682123659880166705099776
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 13 2020

Keywords

Crossrefs

Cf. A007840, A102221, A110083, A193161.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[1/(1 - Sum[x^k/(k k!), {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2
nmax = 16; Assuming[x > 0, CoefficientList[Series[1/(1 + EulerGamma - ExpIntegralEi[x] + Log[x]), {x, 0, nmax}], x]] Range[0, nmax]!^2

Formula

a(n) = (n!)^2 * [x^n] 1 / (1 - Sum_{k>=1} x^k / (k*k!)).
Previous Showing 81-90 of 104 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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