A103213 -id:A103213 - 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

A355171 a(n) = Sum_{k=0..n} binomial(n, k + 1)k!(n + 1)!/(k + 2)!.

Original entry on oeis.org

0, 1, 7, 50, 406, 3804, 41028, 506064, 7084656, 111690720, 1967421600, 38425449600, 825970435200, 19404363283200, 495012834489600, 13632039812966400, 403120633444300800, 12740557701389414400, 428546132879432601600, 15284163618598275072000, 576073025410937628672000

Offset: 0

Peter Luschny, Jun 22 2022

nonn

Cf. A066667, A001705, A103213.

a := n -> n*(n + 1)!*hypergeom([1, 1, 1 - n], [2, 3], -1) / 2;
seq(simplify(a(n)), n = 0..20);

a[n_] := n * (n + 1)! * HypergeometricPFQ[{1, 1, 1 - n}, {2, 3}, -1]/2; Array[a, 21, 0] (* Amiram Eldar, Jun 22 2022 *)

from math import comb, factorial
def A355171(n):
    f = factorial(n+1)
    return sum(f*comb(n,k+1)//(k+2)//(k+1) for k in range(n+1)) # Chai Wah Wu, Jun 22 2022

a(n) = n*(n + 1)!*hypergeom([1, 1, 1 - n], [2, 3], -1) / 2.
a(n) = Sum_{k=0..n} (-1)^(k+1)*k!*A066667(n, k + 1).
E.g.f.: log((1 - x) / (1 - 2*x)) / (1 - x)^2. - Mélika Tebni, Jun 23 2022
a(n) ~ 2^(n+2) * (n-1)!. - Vaclav Kotesovec, Feb 17 2024

A087751 Weighted sum of the harmonic numbers.

Original entry on oeis.org

0, 1, 7, 56, 538, 6124, 81048, 1226112, 20902992, 396857376, 8308373760, 190212376320, 4728556327680, 126865966625280, 3654264347274240, 112484501485977600, 3685202487258163200, 128039255560187596800

Offset: 0

Nicholas C. Singer (nsinger2(AT)cox.net), Oct 02 2003

Cf. A103213, A068102.

H(n)=sum(j=1,n,1/j); a(n)=n!*sum(j=1,n,binomial(n,j)*H(j))

a(n) = 2*n*a(n-1) + (n-1)!*(2^n-1); a(0)=0, a(1)=1. a(n)=n! * sum(j=1, n, binomial(n, j)*H(j)), where H(j)=sum(k=1, j, 1/k).
E.g.f.: log((2*x-1)/(x-1))/(2*x-1). a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*2^(n-k)*binomial(n, k)/k. a(n) = n!*Sum_{k=1..n} 2^(n-k)*(2^k-1)/k. - Vladeta Jovovic, Aug 12 2005
a(n) ~ n! * log(n) * 2^n * (1 + (gamma-log(2))/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 03 2022

A336244 Triangle read by rows: row n gives coefficients T(n,k), in descending powers of m, of a polynomial Q_n(m) (of degree n - 1) in an expression for the number of subdivisions A(m,n) of a grid with two rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 2, 1, 12, 29, 6, 1, 22, 131, 206, 24, 1, 35, 385, 1525, 1774, 120, 1, 51, 895, 6585, 19624, 18204, 720, 1, 70, 1792, 21070, 117019, 281260, 218868, 5040, 1, 92, 3234, 55496, 492849, 2210348, 4483436, 3036144, 40320

Offset: 1

Michel Marcus and Petros Hadjicostas, Jul 14 2020

Let P_(m,n) denote a grid with 2 rows that has m points in the top row and n points in the bottom, aligned at the left, and let the bottom left point be at the origin.
For m > n, the number of subdivisions of P_(m,n) is given by A(m,n) = 2^(m-2)/(n-1)!*Q_n(m), where Q_n(m) is some monic polynomial of degree n-1. See Theorem 2, p. 6, in Robeva and Sun (2020).
By symmetry, A(m,n) = A(n,m). For more information and formulas, see A059576.

			Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins
  1;
  1,  1;
  1,  5,   2;
  1, 12,  29,   6;
  1, 22, 131, 206, 24;
  ...
Q_3(m) = m^2 + 5*m + 2.

Elina Robeva and Melinda Sun, Bimonotone Subdivisions of Point Configurations in the Plane, arXiv:2007.00877 [math.CO], 2020. See Table 2, p. 5.
Wikipedia, Faulhaber's formula.

Cf. A059576, A103213, A336245.

# We assume the rows indexed by the degree of the polynomials, n = 0,1,2,...
A336244row := proc(n) local p, k, s, b; p := 1;
b := n -> bernoulli(n, x+1) - bernoulli(n, 1);
for k from 1 to n-1 do
  s := p + add(coeff(p, x, i-1)*b(i)/i, i=1..k-1);
  p := b(k) + k*s od;
seq(coeff(p, x, n-i), i=1..n) end:
seq(A336244row(n), n=0..9); # Peter Luschny, Jul 15 2020

b[n_] := BernoulliB[n, x + 1] - BernoulliB[n, 1]; b[1] = x;
row[n_] := Module[{p = 1, s}, Do[s = p + Sum[Coefficient[p, x, i-1] b[i]/i, {i, 1, k-1}]; p = b[k] + k s, {k, 1, n-1}]; CoefficientList[p, x] // Reverse];
row /@ Range[9] // Flatten (* Jean-François Alcover, Aug 21 2020, after Peter Luschny *)

polf(n) = if (n==0, return(m)); my(p=bernpol(n+1,m)); (subst(p, m, m+1) - subst(p, m, 0))/(n+1);  \\ Faulhaber
tabl(nn) = {my(p = 1, q); for (n=1, nn, if (n==1, q = p, q = (n-1)*(p + polf(n-2) + sum(i=0, n-3, polcoef(p, i, m)*polf(i)))); print(Vec(q)); p = q;);}

A(m,n) = (2^(m-2)/(n-1)!) * Sum_{k=1..n} T(n,k)*m^(n-k).
A(m,n) = (2^(m-2)/(n-1)!) * Q_n(m) = A059576(m-1, n-1) (provided the latter is viewed as a square array rather than a triangle).
A(m,n) = (2^(m-2)/(n-2)!) * (Q_(n-1)(m) + Sum_{i=1..m} Q_(n-1)(i)).
A(m,n) = 2*(A(m,n-1) + A(m-1,n) - A(m-1,n-1)) for m > n.
T(n, 1) = 1 and T(n, n) = (n - 1)!.
Conjectures:
(a) T(n,2) = (n - 1)*(3*n - 4)/2.
(b) T(n,3) = (n - 2)*(n - 1)*(27*n^2 - 97*n + 72)/24.
(c) T(n,4) = (n - 3)*(n - 2)*(n - 1)^2*(27*n^2 - 156*n + 208)/48.
(d) T(n, n - 1) = (n - 1)!*Sum_{k=1..n-1} binomial(n-1, k)/k = A103213(n-1).

A355257 Array read by ascending antidiagonals. A(n, k) = k! * [x^k] log((1 - x) / (1 - 2*x)) / (1 - x)^n, for 0 <= k <= n.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 5, 14, 0, 1, 7, 29, 90, 0, 1, 9, 50, 206, 744, 0, 1, 11, 77, 406, 1774, 7560, 0, 1, 13, 110, 714, 3804, 18204, 91440, 0, 1, 15, 149, 1154, 7374, 41028, 218868, 1285200, 0, 1, 17, 194, 1750, 13144, 85272, 506064, 3036144, 20603520

Offset: 0

Peter Luschny and Mélika Tebni, Jul 01 2022

Conjecture: For p prime, A(n, p) == -1 (mod p) for n >= 0.
Conjecture: Let n >= 0, k >= 1 and k != 4. Then k divides A(n, k) if and only if k is not prime.
From Mélika Tebni, Jul 04 2022: (Start)
Conjecture: The polynomials of A355259 generate the k+1 column of this array.
Conjecture: For p prime and n even, (A(n, p) / (p - 1)) == 1 (mod p). (End)

			Table A(n, k) begins:
  [0] 0, 1,  3,  14,   90,   744,   7560,    91440,   1285200, ... A029767
  [1] 0, 1,  5,  29,  206,  1774,  18204,   218868,   3036144, ... A103213
  [2] 0, 1,  7,  50,  406,  3804,  41028,   506064,   7084656, ... A355171
  [3] 0, 1,  9,  77,  714,  7374,  85272,  1102968,  15908400, ... A355372
  [4] 0, 1, 11, 110, 1154, 13144, 164136,  2251920,  33923760, ... A355407
  [5] 0, 1, 13, 149, 1750, 21894, 295500,  4320420,  68487120, ... A355414
  [6] 0, 1, 15, 194, 2526, 34524, 502644,  7838928, 131198544, ...
  [7] 0, 1, 17, 245, 3506, 52054, 814968, 13543704, 239548176, ...

Cf. A029767, A103213, A355171, A355259, A355372, A355407, A355414.

egf := n -> log((1 - x)/(1 - 2*x))/(1 - x)^n:
ser := n -> series(egf(n), x, 22):
row := n -> seq(k!*coeff(ser(n), x, k), k = 0..8):
seq(print(row(n)), n = 0..8);
# Alternative:
A := (n, k) -> add(k!*binomial(k + n - 1, k - j - 1)/(j + 1), j = 0..k-1):
seq(print(seq(A(n, k), k = 0..8)), n = 0..7);

A[0, 0] = 0; A[n_, k_] := k! * Binomial[n+k-1, k - 1] * HypergeometricPFQ[{1 - k, 1, 1}, {2, n + 1}, -1];
Table[A[n, k], {n, 0, 8}, {k, 0, 8}] // TableForm

A(n, k) = k!*Sum_{j=0..k-1} binomial(k + n - 1, k - j - 1) / (j + 1).
A(n, k) = k!*Sum_{j=1..k} binomial(n + k - j - 1, n - 1)*(2^j - 1) / j.
A(n, k) = k!*binomial(n + k - 1, k - 1)*hypergeom([1, 1, 1 - k], [2, n + 1], -1) except for A(0, 0) = 0.

cp's OEIS Frontend

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

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A355171 a(n) = Sum_{k=0..n} binomial(n, k + 1)*k!*(n + 1)!/(k + 2)!.

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A087751 Weighted sum of the harmonic numbers.

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A336244 Triangle read by rows: row n gives coefficients T(n,k), in descending powers of m, of a polynomial Q_n(m) (of degree n - 1) in an expression for the number of subdivisions A(m,n) of a grid with two rows.

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A355257 Array read by ascending antidiagonals. A(n, k) = k! * [x^k] log((1 - x) / (1 - 2*x)) / (1 - x)^n, for 0 <= k <= n.

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A294117 a(n) = (n!)^2 * Sum_{k=1..n} binomial(n,k) / k^2.

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A355171 a(n) = Sum_{k=0..n} binomial(n, k + 1)k!(n + 1)!/(k + 2)!.