A134095 -id:A134095 - cp's OEIS frontend

A063170 Schenker sums with n-th term.

1, 2, 10, 78, 824, 10970, 176112, 3309110, 71219584, 1727242866, 46602156800, 1384438376222, 44902138752000, 1578690429731402, 59805147699103744, 2428475127395631750, 105224992014096760832, 4845866591896268695010, 236356356027029797011456

Offset: 0

Marijke van Gans (marijke(AT)maxwellian.demon.co.uk)

Urn, n balls, with replacement: how many selections if we stop after a ball is chosen that was chosen already? Expected value is a(n)/n^n.
Conjectures: The exponent in the power of 2 in the prime factorization of a(n) (its 2-adic valuation) equals 1 if n is odd and equals n - A000120(n) if n is even. - Gerald McGarvey, Nov 17 2007, Jun 29 2012
Amdeberhan, Callan, and Moll (2012) have proved McGarvey's conjectures. - Jonathan Sondow, Jul 16 2012
a(n), for n >= 1, is the number of colored labeled mappings from n points to themselves, where each component is one of two colors. - Steven Finch, Nov 28 2021

			a(4) = (1*2*3*4) + 4*(2*3*4) + 4*4*(3*4) + 4*4*4*(4) + 4*4*4*4.
G.f. = 1 + 2*x + 10*x^2 + 78*x^3 + 824*x^4 + 10970*x^5 + 176112*x^6 + ...

D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, Addison-Wesley, p. 123, Exercise Section 1.2.11.3 18.

G. C. Greubel, Table of n, a(n) for n = 0..385
Max Alekseyev, Recursion for A063170, answer to question on MathOverflow (2025).
T. Amdeberhan, D. Callan, and V. Moll, p-adic analysis and combinatorics of truncated exponential sums, preprint, 2012.
T. Amdeberhan, D. Callan and V. Moll, Valuations and combinatorics of truncated exponential sums, INTEGERS 13 (2013), #A21.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7.
David M. Smith and Geoffrey Smith, Tight Bounds on Information Leakage from Repeated Independent Runs, 2017 IEEE 30th Computer Security Foundations Symposium (CSF).
Marijke van Gans, Schenker sums
Eric Weisstein, Exponential Sum Function.

Cf. A000312, A134095, A090878, A036505, A120266, A214402, A219546 (Schenker primes).

seq(simplify(GAMMA(n+1,n)*exp(n)),n=0..20); # Vladeta Jovovic, Jul 21 2005

a[n_] := Round[ Gamma[n+1, n]*Exp[n]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 16 2012, after Vladeta Jovovic *)
a[ n_] := If[ n < 1, Boole[n == 0], n! Sum[ n^k / k!, {k, 0, n}]]; (* Michael Somos, Jun 05 2014 *)
a[ n_] := If[ n < 0, 0, n! Normal[ Exp[x] + x O[x]^n] /. x -> n]; (* Michael Somos, Jun 05 2014 *)

{a(n) = if( n<0, 0, n! * sum( k=0, n, n^k / k!))};

{a(n) = sum( k=0, n, binomial(n, k) * k^k * (n - k)^(n - k))}; /* Michael Somos, Jun 09 2004 */

for(n=0,17,print1(round(intnum(x=0,[oo,1],exp(-x)*(n+x)^n)),", ")) \\ Gerald McGarvey, Nov 17 2007

from math import comb
def A063170(n): return (sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n) + (n**n<<1) if n else 1 # Chai Wah Wu, Apr 26 2023

10 for N=1 to 42: T=N^N: S=T
20 for K=N to 1 step -1: T/=N: T*=K: S+=T: next K
30 print N,S: next N

a(n) = Sum_{k=0..n} n^k n!/k!.
a(n)/n! = Sum_{k=0..n} n^k/k!. (First n+1 terms of e^n power series.)
a(n) = A063169(n) + n^n.
E.g.f.: 1/(1-T)^2, where T=T(x) is Euler's tree function (see A000169).
E.g.f.: 1 / (1 - F), where F = F(x) is the e.g.f. of A003308. - Michael Somos, May 27 2012
a(n) = Sum_{k=0..n} binomial(n,k)*(n+k)^k*(-k)^(n-k). - Vladeta Jovovic, Oct 11 2007
Asymptotics of the coefficients: sqrt(Pi*n/2)*n^n. - N-E. Fahssi, Jan 25 2008
a(n) = A120266(n)*A214402(n) for n > 0. - Jonathan Sondow, Jul 16 2012
a(n) = Integral_{0..oo} exp(-x) * (n + x)^n dx. - Michael Somos, May 18 2004
a(n) = Integral_{0..oo} exp(-x)*(1+x/n)^n dx * n^n = A090878(n)/A036505(n-1) * n^n. - Gerald McGarvey, Nov 17 2007
EXP-CONV transform of A000312. - Tilman Neumann, Dec 13 2008
a(n) = n! * [x^n] exp(n*x)/(1 - x). - Ilya Gutkovskiy, Sep 23 2017
a(n) = (n+1)! - Sum_{k=0..n-1} binomial(n, k)*a(k)*(-k)^(n-k) for n > 0 with a(0) = 1 (see Max Alekseyev link). - Mikhail Kurkov, Jan 14 2025

A060435 Number of functions f: {1,2,...,n} -> {1,2,...,n} with even cycles only.

Original entry on oeis.org

1, 0, 1, 6, 57, 680, 9945, 172032, 3438673, 78003648, 1980083025, 55616359040, 1712630427849, 57375166877184, 2077563829893097, 80859304977696000, 3366275257190794785, 149270897223530835968, 7024011523121427204897, 349574012216588890718208

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

E.g.f. equals the square-root of the e.g.f. of A134095. - Paul D. Hanna, Oct 11 2007

			E.g.f. A(x) = 1 + 0*x + 1*x^2/2! + 6*x^3/3! + 57*x^4/4! + 680*x^5/5! +...
The formula A(x) = 1/sqrt(1 - LambertW(-x)^2 ) is illustrated by:
A(x) = 1/sqrt(1 - (x+ x^2+ 3^2*x^3/3!+ 4^3*x^4/4!+ 5^4*x^5/5! +...)^2).

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A134095.

Column k=2 of A246609.

t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Range[0, 20]! CoefficientList[Series[(1/(1 - t^2))^(1/2), {x, 0, 20}], x]  (* Geoffrey Critzer, Dec 07 2011 *)

{a(n)=local(LambertW=sum(k=0,n,(-x)^(k+1)*(k+1)^k/(k+1)!) +x*O(x^n)); n!*polcoeff(1/sqrt(1-subst(LambertW,x,-x)^2),n)} \\ Paul D. Hanna, Oct 11 2007

E.g.f.: 1/sqrt(1-(LambertW(-x))^2). a(n)=(n-1)!*Sum_{k=0..floor((n-2)/2)} (k+1)/2^(2*k+1)*binomial(2*k+2, k+1)*n^(n-2-2*k)/(n-2-2*k)!.
A134095(n) = Sum_{k=0..n} C(n,k) * a(n-k) * a(k) with a(0)=1 and a(1)=0 where A134095(n) = Sum_{k=0..n} C(n,k) * (n-k)^k * k^(n-k). - Paul D. Hanna, Oct 11 2007
a(n) ~ n! * 2^(3/4)*Gamma(3/4)*exp(n)/(4*Pi*n^(3/4)) * (1- 5*Pi/ (24*Gamma(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Sep 24 2013

More terms from Alois P. Heinz, Aug 26 2014

A277458 Expansion of e.g.f. -1/(1-LambertW(-x)).

Original entry on oeis.org

-1, 1, 0, 3, 16, 165, 2016, 30415, 539904, 11049129, 256038400, 6627314331, 189517916160, 5933803272397, 201893195083776, 7417376809230375, 292648536838045696, 12341039738944113105, 553942486234823786496, 26369048375194607316019, 1326864458454400696320000

Offset: 0

Vaclav Kotesovec, Oct 16 2016

G. C. Greubel, Table of n, a(n) for n = 0..385
Jason Saied, Jeffrey Marshall, Namit Anand, and Eleanor G. Rieffel, General protocols for the efficient distillation of indistinguishable photons, arxiv:2404.14217 [quant-ph], Apr 22 2024. See p. 14.

Cf. A000312, A086331, A133297, A134095, A277510, A308506.

seq(n!*add((-1)^(k+1)*k*n^(n-k-1)/(n-k)!, k = 1..n), n = 1..20); # Peter Bala, Jul 23 2021

CoefficientList[Series[-1/(1-LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!

my(x='x+O('x^50)); Vec(serlaplace(-1/(1 - lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017

a(n) ~ n^(n-1) / 4.
a(n) = n!*Sum_{k = 1..n} (-1)^(k+1)*k*n^(n-k-1)/(n-k)! for n >= 1. Cf. A133297. - Peter Bala, Jul 23 2021
a(n) = (-1)^(n+1)*U(1-n, -n, -n) where U is the Kummer U function. - Peter Luschny, Jan 23 2025

A277510 E.g.f.: -1/(1-LambertW(-x))^2.

Original entry on oeis.org

-1, 2, -2, 6, 8, 170, 1872, 29246, 519808, 10698642, 248787200, 6458737142, 185138721792, 5808233422394, 197952647108608, 7283047491096750, 287705410381709312, 12145740050403520034, 545696709922799419392, 25998534614835587104742, 1309210567403228200960000

Offset: 0

Vaclav Kotesovec, Oct 18 2016

sign

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A063170, A134095, A277458, A277490, A308506.

CoefficientList[Series[-1/(1-LambertW[-x])^2, {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^50); Vec(serlaplace(-1/(1 - lambertw(-x))^2)) \\ G. C. Greubel, Nov 08 2017

a(n) ~ n^(n-1) / 4.

A277490 E.g.f.: -1/(1+LambertW(-x)^2).

Original entry on oeis.org

-1, 0, 2, 12, 72, 520, 5040, 67284, 1156736, 23655888, 549676800, 14216252380, 405068387328, 12624364306008, 427599019108352, 15646376279614500, 615155126821355520, 25861820048469628576, 1157706908035331457024, 54977324662490442177708, 2760439046217459138560000

Offset: 0

Vaclav Kotesovec, Oct 17 2016

sign

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A063170, A134095.

CoefficientList[Series[-1/(1+LambertW[-x]^2), {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^50); Vec(serlaplace(-1/(1 + lambertw(-x)^2))) \\ G. C. Greubel, Nov 08 2017

a(n) ~ n^(n-1) / 2.

A292977 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x)/(1 - x).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, -1, 1, 6, 1, -2, 2, 2, 24, 1, -3, 5, -2, 9, 120, 1, -4, 10, -12, 8, 44, 720, 1, -5, 17, -34, 33, 8, 265, 5040, 1, -6, 26, -74, 120, -78, 112, 1854, 40320, 1, -7, 37, -138, 329, -424, 261, 656, 14833, 362880, 1, -8, 50, -232, 744, -1480, 1552, -360, 5504, 133496, 3628800

Offset: 0

Ilya Gutkovskiy, Sep 27 2017

A(n,k) is the k-th inverse binomial transform of A000142 evaluated at n.

Can be considered as extension of the array A089258 to columns with negative indices via A089258(n,k) = A(n,-k) or, vice versa, A(n,k) = A089258(n,-k). - Max Alekseyev, Mar 06 2018

			Square array begins:
n=0:    1,   1,   1,    1,     1,      1,  ...
n=1:    1,   0,  -1,   -2,    -3,     -4,  ...
n=2:    2,   1,   2,    5,    10,     17,  ...
n=3:    6,   2,  -2,  -12,   -34,    -74,  ...
n=4:   24,   9,   8,   33,   120,    329,  ...
n=5:  120,  44,   8,  -78,  -424,  -1480,  ...
...
E.g.f. of column k: A_k(x) = 1 + (1 - k)*x/1! +  (k^2 - 2*k + 2)*x^2/2! + (-k^3 + 3*k^2 - 6*k + 6) x^3/3! + (k^4 - 4*k^3 + 12*k^2 - 24*k + 24)*x^4/4! + ...

N. J. A. Sloane, Transforms

Columns: A000142 (k=0), A000166 (k=1), A000023 (k=2), A010843 (k=3, with offset 0).
Main diagonal: A134095 (absolute values).
Cf. A080955, A089258.

Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
FullSimplify[Table[Function[k, Exp[-k] Gamma[n + 1, -k]][j - n], {j, 0, 10}, {n, 0, j}]] // Flatten

T(n, k) = n! * Sum_{j=0..n} (-k)^j/j!. - Max Alekseyev, Mar 06 2018

E.g.f. of column k: exp(-k*x)/(1 - x).

A351768 a(n) = n! * Sum_{k=0..n} k^(n-k) * (n-k)^k/k!.

Original entry on oeis.org

1, 0, 2, 18, 276, 6260, 190950, 7523082, 371286440, 22356290952, 1608686057610, 136069954606190, 13345029902628732, 1500054487474871484, 191349476316804534638, 27464505325501082617170, 4402551348139824475260240, 783025812197886669354545552

Offset: 0

Seiichi Manyama, Feb 18 2022

nonn

Cf. A006153, A062817, A134095, A351765, A351780.

Join[{1}, Table[n!*Sum[k^(n-k) * (n-k)^k/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Feb 19 2022 *)

a(n) = n!*sum(k=0, n, k^(n-k)*(n-k)^k/k!);

log(a(n)) ~ n *(2*log(n) - log(log(n)) - 2 + (log(log(n)) + log(log(n)-1) + 1)/log(n)). - Vaclav Kotesovec, Feb 19 2022

A302398 a(n) = n! * [x^n] 1/(1 + xexp(nx)).

Original entry on oeis.org

1, -1, -2, 3, 248, 5655, 62064, -3516625, -376936064, -21890186577, -495165203200, 96687112380639, 20607024735783936, 2471270260977141767, 142697263160045590528, -25986252776953159328625, -11860424645318274482077696, -2719428501410438623907546529, -372732332273232481973818294272

Offset: 0

Ilya Gutkovskiy, Apr 07 2018

sign

Cf. A134095, A235328, A302397.

Table[n! SeriesCoefficient[1/(1 + x Exp[n x]), {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n! Sum[(-1)^(n - k) (n (n - k))^k/k!, {k, 0, n}], {n, 18}]]
Join[{1}, Table[Sum[(-1)^k k! (n k)^(n - k) Binomial[n, k], {k, 0, n}], {n, 18}]]

a(n) = n!*Sum_{k=0..n} (-1)^(n-k)*(n*(n-k))^k/k!.

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

A332627 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * k! * k^n.

Original entry on oeis.org

1, 1, 6, 117, 4388, 266065, 23731314, 2923345621, 475364541672, 98623225721601, 25421365316232710, 7969388199705535141, 2985785305877403047820, 1317500933136749853197329, 676266417871227455138941242, 399516621958550611386236160405

Offset: 0

Ilya Gutkovskiy, Apr 23 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A000166, A120485, A134095, A302397, A319392, A332408.

Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] k! k^n, {k, 0, n}], {n, 1, 15}]]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * k! * k^n); \\ Michel Marcus, Apr 24 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x*exp(-x))^k))) \\ Seiichi Manyama, Feb 19 2022

G.f.: Sum_{k>=0} k! * k^k * x^k / (1 + k*x)^(k+1).
a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^n / (n-k)!.
a(n) ~ c * n! * n^n, where c = A072364 = exp(-exp(-1)). - Vaclav Kotesovec, Jul 10 2021
E.g.f.: Sum_{k>=0} (k*x*exp(-x))^k. - Seiichi Manyama, Feb 19 2022

A302581 a(n) = n! * [x^n] -exp(-nx)log(1 - x).

Original entry on oeis.org

0, 1, -3, 20, -186, 2249, -33360, 586172, -11901008, 274098393, -7060189120, 201092672604, -6275340884736, 212915635727313, -7803567334571008, 307245946117223700, -12933084380738398208, 579587518114690731601, -27550568677612746940416, 1384553892443352890245636

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

sign

N. J. A. Sloane, Transforms
Index entries for sequences related to logarithmic numbers

Cf. A002104, A002741, A104150, A134095, A190314.

Table[n! SeriesCoefficient[-Exp[-n x] Log[1 - x], {x, 0, n}], {n, 0, 19}]
Table[Sum[(-n)^(n - k) (k - 1)! Binomial[n, k], {k, 1, n}], {n, 0, 19}]
nmax = 20; CoefficientList[Series[-Log[1 - LambertW[x]]/(1 + LambertW[x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 09 2019 *)

a(n) = Sum_{k=1..n} (-n)^(n-k)*(k-1)!*binomial(n,k).
E.g.f.: -log(1 - LambertW(x))/(1 + LambertW(x)). - Vaclav Kotesovec, Jun 09 2019
a(n) ~ -(-1)^n * log(2) * n^n. - Vaclav Kotesovec, Jun 09 2019

cp's OEIS Frontend

A063170 Schenker sums with n-th term.

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UBASIC

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A060435 Number of functions f: {1,2,...,n} -> {1,2,...,n} with even cycles only.

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A277458 Expansion of e.g.f. -1/(1-LambertW(-x)).

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A277510 E.g.f.: -1/(1-LambertW(-x))^2.

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A277490 E.g.f.: -1/(1+LambertW(-x)^2).

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A292977 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x)/(1 - x).

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A351768 a(n) = n! * Sum_{k=0..n} k^(n-k) * (n-k)^k/k!.

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A302398 a(n) = n! * [x^n] 1/(1 + xexp(nx)).

A302581 a(n) = n! * [x^n] -exp(-nx)log(1 - x).