A046825 -id:A046825 - cp's OEIS frontend

A003149 a(n) = Sum_{k=0..n} k!*(n - k)!.

1, 2, 5, 16, 64, 312, 1812, 12288, 95616, 840960, 8254080, 89441280, 1060369920, 13649610240, 189550368000, 2824077312000, 44927447040000, 760034451456000, 13622700994560000, 257872110354432000, 5140559166898176000, 107637093007589376000, 2361827297364885504000

Offset: 0

N. J. A. Sloane, Henry Gould

From Michael Somos, Feb 14 2002: (Start)
The sequence is the resistance between opposite corners of an (n+1)-dimensional hypercube of unit resistors, multiplied by (n+1)!.
The resistances for n+1 = 1,2,3,... are 1, 1, 5/6, 2/3, 8/15, 13/30, 151/420, 32/105, 83/315, 73/315, 1433/6930, ... (see A046878/A046879). (End)
Number of {12,21*,2*1}-avoiding signed permutations in the hyperoctahedral group.
a(n) is the sum of the reciprocals of the binomial coefficients C(n,k), multiplied by n!; example: a(4) = 4!*(1/1 + 1/4 + 1/6 + 1/4 + 1/1) = 64. - Philippe Deléham, May 12 2005
a(n) is the number of permutations on [n+1] that avoid the pattern 13-2|. The absence of a dash between 1 and 3 means the "1" and "3" must be consecutive in the permutation; the vertical bar means the "2" must occur at the end of the permutation. For example, 24153 fails to avoid this pattern: 243 is an offending subpermutation. - David Callan, Nov 02 2005
n!/a(n) is the probability that a random walk on an (n+1)-dimensional hypercube will visit the diagonally opposite vertex before it returns to its starting point. 2^n*a(n)/n! is the expected length of a random walk from one vertex of an (n+1)-dimensional hypercube to the diagonally opposite vertex (a walk which may include one or more passes through the starting point). These "random walk" examples are solutions to IBM's "Ponder This" puzzle for April, 2006. - Graeme McRae, Apr 02 2006
a(n) is the number of strong fixed points in all permutations of {1,2,...,n+1} (a permutation p of {1,2,...,n} is said to have j as a strong fixed point (splitter) if p(k)j for k>j). Example: a(2)=5 because the permutations of {1,2,3}, with marked strong fixed points, are: 1'2'3', 1'32, 312, 213', 231 and 321. - Emeric Deutsch, Oct 28 2008
Coefficients in the asymptotic expansion of exp(-2*x)*Ei(x)^2 for x -> inf, where Ei(x) is the exponential integral. - Vladimir Reshetnikov, Apr 24 2016

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (1.1.11 b, p.342).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Volume 1 (1986), p. 49. [From Emeric Deutsch, Oct 28 2008]

T. D. Noe, Table of n, a(n) for n = 0..100
Joerg Arndt, Generating Random Permutations, PhD thesis, Australian National University, Canberra, Australia, (2010).
Fred Curtis, Resistance-network Problems.
Bakir Farhi, On a curious integer sequence, arXiv:2204.10136 [math.NT], 2022.
Todd Feil, Gary Kennedy and David Callan, Problem E3467, Amer. Math. Monthly, 100 (1993), 800-801. [From _Emeric Deutsch_, Oct 28 2008]
Henry W. Gould, Letter to N. J. A. Sloane, Nov 1973, and various attachments.
IBM, "Ponder This" puzzle for April, 2006
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
F. Nedemeyer and Y. Smorodinsky, Resistance in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.
R. Sprugnoli, Moments of Reciprocals of Binomial Coefficients, Journal of Integer Sequences, 14 (2011), #11.7.8.
V. Strehl, The average number of splitters in a random permutation [Unpublished; included here with the author's permission.]
B. Sury, Sum of the reciprocals of the binomial coefficients, Europ. J. Comb., 14 (1993), 351-353.
Eric Weisstein's World of Mathematics, Incomplete Beta Function.
Eric Weisstein's World of Mathematics, Lerch Transcendent.
Index to sequences related to resistances.

Cf. A046825, A046878, A046879.
Cf. A052186, A006932, A145878. - Emeric Deutsch, Oct 28 2008
Cf. A324495, A324496, A324497 (problem similar to the random walks on the hypercube).

F:=Factorial;; List([0..20], n-> Sum([0..n], k-> F(k)*F(n-k)) ); # G. C. Greubel, Dec 29 2019

F:=Factorial; [ (&+[F(k)*F(n-k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 29 2019

seq( add(k!*(n-k)!, k=0..n), n=0..20); # G. C. Greubel, Dec 29 2019
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n+1,
      ((3*n+1)*a(n-1)-n^2*a(n-2))/2)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Aug 08 2025

Table[Sum[k!(n-k)!,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Mar 28 2012 *)
Table[(n+1)!/2^n*Sum[2^k/(k+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 27 2012 *)
Round@Table[-2 (n+1)! Re[LerchPhi[2, 1, n+2]], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 12 2015 *)
Table[(n+1)!*Sum[Binomial[n+1, 2*j+1]/(2*j+1), {j, 0, n}]/2^n, {n, 0, 20}] (* Vaclav Kotesovec, Dec 04 2015 *)
Series[Exp[-2x] ExpIntegralEi[x]^2, {x, Infinity, 20}][[3]] (* Vladimir Reshetnikov, Apr 24 2016 *)
Table[2*(-1)^n * Sum[(2^k - 1) * StirlingS1[n, k] * BernoulliB[k], {k, 0, n}], {n, 1, 25}] (* Vaclav Kotesovec, Oct 04 2022 *)

PARI
```
a(n)=sum(k=0,n,k!*(n-k)!)
```

a(n)=if(n<0,0,(n+1)!*polcoeff(log(1-x+x^2*O(x^n))/(x/2-1),n+1))

a(n) = my(A = 1, B = 1); for(k=1, n, B *= k; A = (n-k+1)*A + B); A \\ Mikhail Kurkov, Aug 08 2025

def a(n: int) -> int:
    if n < 2: return n + 1
    app, ap = 1, 2
    for i in range(2, n + 1):
        app, ap = ap, ((3 * i + 1) * ap - (i * i) * app) >> 1
    return ap
print([a(n) for n in range(23)])  # Peter Luschny, Aug 08 2025

f=factorial; [sum(f(k)*f(n-k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Dec 29 2019

a(n) = n! + ((n+1)/2)*a(n-1), n >= 1. - Leroy Quet, Sep 06 2002
a(n) = ((3n+1)*a(n-1) - n^2*a(n-2))/2, n >= 2. - David W. Wilson, Sep 06 2002; corrected by N. Sato, Jan 27 2010
G.f.: (Sum_{k>=0} k!*x^k)^2. - Vladeta Jovovic, Aug 30 2002
E.g.f: log(1-x)/(x/2 - 1) if offset 1.
Convolution of A000142 [factorial numbers] with itself. - Ross La Haye, Oct 29 2004
a(n) = Sum_{k=0..n+1} k*A145878(n+1,k). - Emeric Deutsch, Oct 28 2008
a(n) = A084938(n+2,2). - Philippe Deléham, Dec 17 2008
a(n) = 2*Integral_{t=0..oo} Ei(t)*exp(-2*t)*t^(n+1) where Ei is the exponential integral function. - Groux Roland, Dec 09 2010
Empirical: a(n-1) = 2^(-n)*(A103213(n) + n!*H(n)) with H(n) harmonic number of order n. - Groux Roland, Dec 18 2010; offset fixed by Vladimir Reshetnikov, Apr 24 2016
O.g.f.: 1/(1-I(x))^2 where I(x) is o.g.f. for A003319. - Geoffrey Critzer, Apr 27 2012
a(n) ~ 2*n!. - Vaclav Kotesovec, Oct 04 2012
a(n) = (n+1)!/2^n * Sum_{k=0..n} 2^k/(k+1). - Vaclav Kotesovec, Oct 27 2012
E.g.f.: 2/((x-1)*(x-2)) + 2*x/(x-2)^2*G(0) where G(k) = 1 + x*(2*k+1)/(2*(k+1) - 4*x*(k+1)^2/(2*x*(k+1) + (2*k+3)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 14 2012
a(n) = 2 * n! * (1 + Sum_{k>=1} A005649(k-1)/n^k). - Vaclav Kotesovec, Aug 01 2015
From Vladimir Reshetnikov, Nov 12 2015: (Start)
a(n) = -(n+1)!*Re(Beta(2; n+2, 0))/2^(n+1), where Beta(z; a, b) is the incomplete Beta function.
a(n) = -2*(n+1)!*Re(LerchPhi(2, 1, n+2)), where LerchPhi(z, s, a) is the Lerch transcendent. (End)
a(n) = (n+1)!*(H(n+1) + (n+1)*hypergeom([1, 1, -n], [2, 2], -1))/2^(n+1), where H(n) is the harmonic number. - Vladimir Reshetnikov, Apr 24 2016
Expansion of square of continued fraction 1/(1 - x/(1 - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - ...))))))). - Ilya Gutkovskiy, Apr 19 2017
a(n) = Sum_{k=0..n+1} (-1)^(n-k)*A226158(k)*Stirling1(n+1, k). - Mélika Tebni, Feb 22 2022
E.g.f.: x/((1-x)*(2-x))-(2*log(1-x))/(2-x)^2+1/(1-x). - Vladimir Kruchinin, Dec 17 2022

More terms from Michel ten Voorde, Apr 11 2001

A046878 Numerator of (1/n)*Sum_{k=0..n-1} 1/binomial(n-1,k) for n>0 else 0.

Original entry on oeis.org

0, 1, 1, 5, 2, 8, 13, 151, 32, 83, 73, 1433, 647, 15341, 28211, 10447, 1216, 19345, 18181, 651745, 1542158, 1463914, 2786599, 122289917, 29229544, 140001721, 134354573, 774885169, 745984697, 41711914513, 80530073893, 4825521853483

Offset: 0

N. J. A. Sloane

a(n) is also the numerator of (1/2^n)*Sum_{k=1..n} 2^k/k. - Groux Roland, Jan 13 2009

			Rational sequence starts: 0, 1, 1, 5/6, 2/3, 8/15, 13/30, 151/420, 32/105,...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle

See A046825, the main entry for this sequence. Cf. A046879.

a := n -> -2*LerchPhi(2,1,n+1)-I*Pi/2^n:
seq(numer(simplify(a(n))),n=0..31); # Peter Luschny, Nov 20 2015

a[0] = 0; a[n_] := (1/n) Sum[1/Binomial[n-1, k], {k, 0, n-1}] // Numerator; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Sep 28 2016 *)

a(n):=if n=0 then 0 else num((-1)^(n-1)/(n-1)!*sum(2^k*bern(k)*(stirling1(n-1,k)),k,0,n-1)); /* Vladimir Kruchinin, Nov 20 2015 */

vector(40, n, n--; numerator((1/2^n)*sum(k=1, n, 2^k/k))) \\ Altug Alkan, Nov 20 2015

a(n) = numerator((-1)^(n-1)/(n-1)!*Sum_{k=0..n-1} 2^k*bernoulli(k)* stirling1(n-1,k)), n>0, a(0)=0. - Vladimir Kruchinin, Nov 20 2015

a(n) = numerator(-2*LerchPhi(2,1,n+1)-i*Pi/2^n). - Peter Luschny, Nov 20 2015

A046879 Denominator of (1/n)*Sum_{k=0..n-1} 1/binomial(n-1,k) for n>0 else 1.

Original entry on oeis.org

1, 1, 1, 6, 3, 15, 30, 420, 105, 315, 315, 6930, 3465, 90090, 180180, 72072, 9009, 153153, 153153, 5819814, 14549535, 14549535, 29099070, 1338557220, 334639305, 1673196525, 1673196525, 10039179150, 10039179150, 582272390700, 1164544781400

Offset: 0

N. J. A. Sloane

For n>=1 a(n) is the denominator of (1/2^n)*Sum_{k=1..n} 2^k/k. - Groux Roland, Jan 13 2009

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle

See A046825, the main entry for this sequence. Cf. A046878.

a := n -> -2*LerchPhi(2,1,n+1)-I*Pi/2^n:
seq(denom(simplify(a(n))),n=0..30); # Peter Luschny, Nov 20 2015

Denominator[Simplify[-2*LerchPhi[2, 1, # + 1] - I*Pi/2^#]] & /@
Range[0, 100] (* Julien Kluge, Jul 21 2016 *)

a(n):=if n=0 then 1 else denom((-1)^(n-1)/(n-1)!*sum(2^k*bern(k)*(stirling1(n-1,k)),k,0,n-1)); /* Vladimir Kruchinin, Nov 20 2015 */

vector(30, n, n--; denominator((1/2^n)*sum(k=1, n, 2^k/k))) \\ Altug Alkan, Nov 20 2015

a(n) = denominator((-1)^(n-1)/(n-1)!*Sum_{k=0..n-1} 2^k*bern(k) * stirling1(n-1,k)), n>0, a(0)=1. - Vladimir Kruchinin, Nov 20 2015

a(n) = denominator(-2*LerchPhi(2,1,n+1)-i*Pi/2^n). - Peter Luschny, Nov 20 2015

A100516 Numerator of Sum_{k=0..n} 1/binomial(n,k)^2.

Original entry on oeis.org

1, 2, 9, 20, 155, 21, 7441, 3224, 5697, 3575, 28523, 27183, 70357417, 4661447, 386395, 8959408, 10028928779, 525966759, 1476346738309, 35051863075, 847581175, 709068173, 62385202783, 20340152122, 119483756745025, 4418168441921, 311960929172031

Offset: 0

N. J. A. Sloane, Nov 25 2004

			1, 2, 9/4, 20/9, 155/72, 21/10, 7441/3600, 3224/1575, 5697/2800, 3575/1764, 28523/14112, 27183/13475, 70357417/34927200, 4661447/2316600, ... = A100516/A100517

H. W. Gould, Combinatorial Identities, Morgantown, 1972, p. 50, formula (5.2).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A046825, A100517, A100518.

[Numerator( (&+[1/Binomial(n,k)^2: k in [0..n]]) ): n in [0..40]]; // G. C. Greubel, Jun 24 2022

Table[3*(n+1)^2/((n+2)*(2*n+3)*CatalanNumber[n+1])*Sum[((k+ 1)/k)*CatalanNumber[k], {k,n+1}], {n,0,40}]//Numerator (* G. C. Greubel, Jun 24 2022 *)

a(n) = numerator(sum(k=0, n, 1/binomial(n,k)^2)); \\ Michel Marcus, Jun 24 2022

[numerator(sum(1/binomial(n,k)^2 for k in (0..n))) for n in (0..40)] # G. C. Greubel, Jun 24 2022

a(n) = numerator( 3*(n+1)^2/((n+2)*(2*n+3)*Catalan(n+1)) * Sum_{k=1..n+1} binomial(2*k, k)/k ). - G. C. Greubel, Jun 24 2022

A051389 Number of resistance values that can be constructed using exactly n 1-ohm resistors in series or parallel but not with fewer resistors.

Original entry on oeis.org

1, 2, 4, 8, 20, 42, 102, 250, 610, 1486, 3710, 9228, 23050, 57718, 145288, 365820, 922194, 2327914, 5885800, 14890796, 37701452, 95550472, 242325118, 614869792, 1561228066, 3966071764, 10080113232, 25630109268, 65194419268, 165890640468

Offset: 1

Hugo van der Sanden

If x and y require xn and yn resistors respectively, then (x+y) and 1/(1/x + 1/y) require no more than (xn+yn). Inspired by a sci.math posting by Miguel A. Lerma (lerma(AT)math.nwu.edu).
Let A(n) be the set of resistances equivalent to a network of n 1-ohm resistors using only series and parallel combinations. Then A048211(n) = card(A(n)). Let L(n) be the set of resistances that first appear in A(n), i.e. L(n) = A(n) \ (A(1) U ... U A(n-1)). Then a(n) = card(L(n)). - Antoine Mathys, Nov 22 2024
If a resistance is equivalent to a n-resistor circuit, then it is equivalent to a 4n-resistor circuit. There is therefore no upper bound on the size of the networks to which it is equivalent. - Antoine Mathys, Nov 22 2024

			The a(1) = 1 resistance value is 1 ohm.
The a(2) = 2 resistance values are {1/2, 2}.
The a(3) = 4 resistance values are {1/3, 2/3, 3/2, 3}.
The a(4) = 8 resistance values are {1/4, 2/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4}.
The a(5) = 20 resistance values are {1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5, 5/6, 6/7, 7/6, 6/5, 5/4, 7/5, 8/5, 7/4, 7/3, 8/3, 7/2, 5}.
E.g. 6/5 is made from two resistors in series in parallel with three resistors in series, since 6/5 = 1/(1/2 + 1/3). It cannot be obtained using fewer resistors.

Miguel A. Lerma, resistors, post in the newsgroup sci.math, Nov 5 1999.
Index to sequences related to resistances.

Cf. A048211, A046825, A153588, A180414, A338197.

a(n) = A153588(n) - A153588(n-1) for n > 1. - Hugo Pfoertner, Nov 04 2020

a(15)-a(21) from Jon E. Schoenfield, Aug 28 2006
Definition corrected by Jon E. Schoenfield, Aug 27 2006
a(22)-a(23) from Graeme McRae, Aug 18 2007
a(24)-a(25) from Antoine Mathys, Mar 20 2017
Definition changed to say "exactly". - N. J. A. Sloane, Nov 07 2020
Definition clarified by Antoine Mathys, Nov 22 2024
a(26)-a(30) from Antoine Mathys, Dec 05 2024

A100518 Numerator of Sum_{k=0..n} 1/binomial(n,k)^3.

Original entry on oeis.org

1, 2, 17, 56, 1759, 1009, 86831, 2322304, 85922, 1144667, 16019198113, 123357293, 21312406359367, 17061774340031, 27741170437991, 182851619022848, 167169857863289, 9857517443932187, 8844183281912559671, 197147246106875452361, 681198614358931646209

Offset: 0

N. J. A. Sloane, Nov 25 2004

			1, 2, 17/8, 56/27, 1759/864, 1009/500, 86831/43200, 2322304/1157625, 85922/42875, 1144667/571536, 16019198113/8001504000, 123357293/61631955, ... = A100518/A100519.

G. C. Greubel, Table of n, a(n) for n = 0..770

Cf. A046825, A046826, A100516, A100517, A100519.

[Numerator( (&+[1/Binomial(n,k)^3: k in [0..n]]) ): n in [0..40]]; // G. C. Greubel, Jun 24 2022

Numerator[Table[Sum[1/Binomial[n,k]^3,{k,0,n}],{n,0,20}]] (* Harvey P. Dale, Sep 28 2012 *)

a(n) = numerator(sum(k=0, n, 1/binomial(n,k)^3)); \\ Michel Marcus, Jun 24 2022

[numerator(sum(1/binomial(n,k)^3 for k in (0..n))) for n in (0..40)] # G. C. Greubel, Jun 24 2022

a(n) = numerator( Sum_{k=0..n} 1/binomial(n,k)^3 ).

cp's OEIS Frontend

A214432 Indices for which A046825, the numerator of sum( 1/binomial(n,i), i=0..n ), sets a new record.

Views

Author

Keywords

Comments

Programs

PARI

A214440 Indices n for which A046825(n) is not larger than A046825(n-1).

Views

Author

Keywords

Crossrefs

Programs

PARI

A214453 Odd indices n for which A046825(n) is not larger than A046825(n-1).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A003149 a(n) = Sum_{k=0..n} k!*(n - k)!.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Sage

Formula

Extensions

A046878 Numerator of (1/n)*Sum_{k=0..n-1} 1/binomial(n-1,k) for n>0 else 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A046879 Denominator of (1/n)*Sum_{k=0..n-1} 1/binomial(n-1,k) for n>0 else 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A100516 Numerator of Sum_{k=0..n} 1/binomial(n,k)^2.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma