A320080 -id:A320080 - cp's OEIS frontend

A006252 Expansion of e.g.f. 1/(1 - log(1+x)).

1, 1, 1, 2, 4, 14, 38, 216, 600, 6240, 9552, 319296, -519312, 28108560, -176474352, 3998454144, -43985078784, 837126163584, -12437000028288, 237195036797184, -4235955315745536, 85886259443020800, -1746536474655406080, 38320721602434017280, -864056965711935974400

Offset: 0

N. J. A. Sloane

sign

From Michael Somos, Mar 04 2004: (Start)
Stirling transform of a(n+1)=[1,2,4,14,38,...] is A000255(n)=[1,3,11,53,309,...].
Stirling transform of 2*a(n)=[2,2,4,8,28,...] is A052849(n)=[2,4,12,48,240,...].
Stirling transform of a(n)=[1,1,2,4,14,38,216,...] is A000142(n)=[1,2,6,24,120,...].
Stirling transform of a(n-1)=[1,1,1,2,4,14,38,...] is A000522(n-1)=[1,2,5,16,65,...].
Stirling transform of a(n-1)=[0,1,1,2,4,14,38,...] is A007526(n-1)=[0,1,4,15,64,...].
(End)
For n > 0: a(n) = sum of n-th row in triangle A048594. - Reinhard Zumkeller, Mar 02 2014
Coefficients in a factorial series representation of the exponential integral: exp(z)*E_1(z) = Sum_{n >= 0} (-1)^n*a(n)/(z)n, where (z)_n denotes the rising factorial z*(z + 1)*...*(z + n) and E_1(z) = Integrate{t = z..inf} exp(-t)/t dt. See Weninger, equation 6.4. - Peter Bala, Feb 12 2019

G. Pólya, Induction and Analogy in Mathematics. Princeton Univ. Press, 1954, p. 9.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Beáta Bényi and Daniel Yaqubi, Mixed coloured permutations, arXiv:1903.07450 [math.CO], 2019.
Takao Komatsu and Amalia Pizarro-Madariaga, Harmonic numbers associated with inversion numbers in terms of determinants, Turkish Journal of Mathematics (2019) Vol. 43, 340-354.
E. J. Weniger, Summation of divergent power series by means of factorial series arXiv:1005.0466v1 [math.NA], 2010.

Column k=1 of A320080.

Cf. A007840.

a006252 0 = 1
a006252 n = sum $ a048594_row n  -- Reinhard Zumkeller, Mar 02 2014

With[{nn=30},CoefficientList[Series[1/(1-Log[1+x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 12 2016 *)

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

{a(n)=local(CF=1+x*O(x^n)); for(k=0, n-1, CF=1/((n-k+1)-(n-k)*x+(n-k+1)^2*x*CF)); n!*polcoeff(1+x/(1-x+x*CF), n, x)} /* Paul D. Hanna, Dec 31 2011 */

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (-1)^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v; \\ Seiichi Manyama, May 22 2022

def A006252_list(len):
    f, R, C = 1, [1], [1]+[0]*len
    for n in (1..len):
        f *= n
        for k in range(n, 0, -1):
            C[k] = -C[k-1]*((k-1)/(k) if k>1 else 1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0]*f)
    return R
print(A006252_list(24)) # Peter Luschny, Feb 21 2016

a(n) = Sum_{k=0..n} k!*stirling1(n, k). - Vladeta Jovovic, Sep 08 2002
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator exp(-x)*d/dx. Row sums of A048594. Cf. A007840. - Peter Bala, Nov 25 2011
E.g.f.: 1/(1-log(1+x)) = 1 + x/(1-x + x/(2-x + 4*x/(3-2*x + 9*x/(4-3*x + 16*x/(5-4*x + 25*x/(6-5*x +...)))))), a continued fraction. - Paul D. Hanna, Dec 31 2011
a(n)/n! ~ -(-1)^n / (n * (log(n))^2) * (1 - 2*(1 + gamma)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 01 2018
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, May 22 2022

A088501 Expansion of e.g.f. 1/(1-2*log(1+x)).

Original entry on oeis.org

1, 2, 6, 28, 172, 1328, 12272, 132480, 1633344, 22663104, 349324608, 5923548288, 109570736256, 2195765044224, 47386235513856, 1095689316882432, 27023900076988416, 708173307424456704, 19649589144733089792

Offset: 0

Vladeta Jovovic, Nov 12 2003

Seiichi Manyama, Table of n, a(n) for n = 0..419

Column k=2 of A320080.

CoefficientList[Series[1/(1-2*Log[1+x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, May 03 2015 *)

a(n) = sum(k=0, n, k!*2^k*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 03 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*log(1+x)))) \\ Seiichi Manyama, Feb 03 2022

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=2*sum(j=1, i, (-1)^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v; \\ Seiichi Manyama, May 22 2022

a(n) = Sum_{k=0..n} Stirling1(n, k)*k!*2^k.
a(n) ~ n! * exp(1/2) / (2 * (exp(1/2)-1)^(n+1)). - Vaclav Kotesovec, May 03 2015
a(0) = 1; a(n) = 2 * Sum_{k=1..n} (-1)^(k-1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, May 22 2022

A335531 Expansion of e.g.f. 1/(1-3*log(1+x)).

Original entry on oeis.org

1, 3, 15, 114, 1152, 14562, 220842, 3907656, 79019496, 1797660000, 45439902288, 1263456328032, 38324061498672, 1259345712721392, 44565940575178992, 1689757622095909248, 68339921117338411776, 2936658673480397537664, 133615257668682429428352, 6417113656859478628233984, 324414161427519766056847104

Offset: 0

Seiichi Manyama, Jun 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..390

Column k=3 of A320080.

Cf. A335530.

a[n_] := Sum[k! * 3^k * StirlingS1[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Jun 12 2020 *)
With[{nn=20},CoefficientList[Series[1/(1-3Log[1+x]),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 02 2021 *)

a(n) = sum(k=0, n, 3^k*k!*stirling(n, k, 1));

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-3*log(1+x))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=3*sum(j=1, i, (-1)^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v; \\ Seiichi Manyama, May 22 2022

a(n) = Sum_{k=0..n} 3^k * k! * Stirling1(n,k).
a(n) ~ n! * exp(1/3) / (3*(exp(1/3)-1)^(n+1)). - Vaclav Kotesovec, Jun 12 2020
a(0) = 1; a(n) = 3 * Sum_{k=1..n} (-1)^(k-1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, May 22 2022

A317172 a(n) = n! * [x^n] 1/(1 - n*log(1 + x)).

Original entry on oeis.org

1, 1, 6, 114, 4168, 248870, 21966768, 2685571560, 434202400896, 89679267601632, 23032451508686400, 7199033431349412576, 2690461258552995849216, 1184680716090974803461072, 606986901206377433194091520, 358023049940533240478842992000, 240858598980174362552808566194176

Offset: 0

Ilya Gutkovskiy, Jul 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A006252, A088501, A094420, A242817, A317171.

Main diagonal of A320080.

Table[n! SeriesCoefficient[1/(1 - n Log[1 + x]), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[StirlingS1[n, k] n^k k!, {k, n}], {n, 16}]]

{a(n) = sum(k=0, n, k!*n^k*stirling(n, k, 1))} \\ Seiichi Manyama, Jun 12 2020

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

a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n + 1/2). - Vaclav Kotesovec, Jul 23 2018

A320079 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k*log(1 - x)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 14, 0, 1, 4, 21, 76, 88, 0, 1, 5, 36, 222, 772, 694, 0, 1, 6, 55, 488, 3132, 9808, 6578, 0, 1, 7, 78, 910, 8824, 55242, 149552, 72792, 0, 1, 8, 105, 1524, 20080, 199456, 1169262, 2660544, 920904, 0, 1, 9, 136, 2366, 39708, 553870, 5410208, 28873800, 54093696, 13109088, 0

Offset: 0

Ilya Gutkovskiy, Oct 05 2018

			E.g.f. of column k: A_k(x) = 1 + k*x/1! + k*(2*k + 1)*x^2/2! + 2*k*(3*k^2 + 3*k + 1)*x^3/3! + 2*k*(12*k^3 + 18*k^2 + 11*k + 3)*x^4/4! + ...
Square array begins:
  1,    1,     1,      1,       1,       1,  ...
  0,    1,     2,      3,       4,       5,  ...
  0,    3,    10,     21,      36,      55,  ...
  0,   14,    76,    222,     488,     910,  ...
  0,   88,   772,   3132,    8824,   20080,  ...
  0,  694,  9808,  55242,  199456,  553870,  ...

Columns k=0..5 give A000007, A007840, A088500, A354263, A354264, A365588.
Main diagonal gives A317171.
Cf. A048594, A048994, A094416, A320080.

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

E.g.f. of column k: 1/(1 + k*log(1 - x)).
A(n,k) = Sum_{j=0..n} |Stirling1(n,j)|*j!*k^j.
A(0,k) = 1; A(n,k) = k * Sum_{j=1..n} (j-1)! * binomial(n,j) * A(n-j,k). - Seiichi Manyama, May 22 2022

A354147 Expansion of e.g.f. 1/(1 - 4 * log(1+x)).

Original entry on oeis.org

1, 4, 28, 296, 4168, 73376, 1550048, 38202048, 1076017344, 34096092672, 1200459182592, 46492497859584, 1964295942558720, 89906908894150656, 4431634108980264960, 234044235939806232576, 13184410813249253031936, 789137065405617987354624

Offset: 0

Seiichi Manyama, May 21 2022

nonn

Column k=4 of A320080.

Cf. A094417, A354240, A354264.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-4*log(1+x))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=4*sum(j=1, i, (-1)^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

a(n) = sum(k=0, n, 4^k*k!*stirling(n, k, 1));

a(0) = 1; a(n) = 4 * Sum_{k=1..n} (-1)^(k-1) * (k-1)! * binomial(n,k) * a(n-k).
a(n) = Sum_{k=0..n} 4^k * k! * Stirling1(n, k).
a(n) ~ n! * exp(1/4) / (4 * (exp(1/4)-1)^(n+1)). - Vaclav Kotesovec, Jun 04 2022

A365604 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x)).

Original entry on oeis.org

1, 5, 45, 610, 11020, 248870, 6744350, 213233400, 7704814200, 313199930400, 14146162064400, 702826758144000, 38093116667766000, 2236695336601458000, 141433354184701746000, 9582086196220281456000, 692463727252196674560000

Offset: 0

Seiichi Manyama, Sep 11 2023

nonn

Column k=5 of A320080.
Cf. A347022, A365601, A365602, A365603.
Cf. A094418, A365588.

a[n_] := Sum[5^k * k! * StirlingS1[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 13 2023 *)
With[{nn=20},CoefficientList[Series[1/(1-5*Log[1+x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 05 2025 *)

a(n) = sum(k=0, n, 5^k*k!*stirling(n, k, 1));

a(n) = Sum_{k=0..n} 5^k * k! * Stirling1(n,k).

a(0) = 1; a(n) = 5 * Sum_{k=1..n} (-1)^(k-1) * (k-1)! * binomial(n,k) * a(n-k).

A334369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - (k-1)log(1 + x))/(1 - klog(1 + x)).

Original entry on oeis.org

1, 1, 1, 1, 1, -1, 1, 1, 1, 2, 1, 1, 3, 2, -6, 1, 1, 5, 14, 4, 24, 1, 1, 7, 38, 86, 14, -120, 1, 1, 9, 74, 384, 664, 38, 720, 1, 1, 11, 122, 1042, 4854, 6136, 216, -5040, 1, 1, 13, 182, 2204, 18344, 73614, 66240, 600, 40320, 1, 1, 15, 254, 4014, 49774, 387512, 1302552, 816672, 6240, -362880

Offset: 0

Seiichi Manyama, Jun 12 2020

			Square array begins:
   1,  1,   1,    1,     1,     1, ...
   1,  1,   1,    1,     1,     1, ...
  -1,  1,   3,    5,     7,     9, ...
   2,  2,  14,   38,    74,   122, ...
  -6,  4,  86,  384,  1042,  2204, ...
  24, 14, 664, 4854, 18344, 49774, ...

Columns k=1..3 give A006252, A308878, A335530.
Main diagonal gives A335529.
Cf. A320080.

T[0, k_] = 1; T[n_, k_] := Sum[If[k == 0 && j <= 1, 1, k^(j - 1)] * j! * StirlingS1[n, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)

T(0,k)=1 and T(n,k) = Sum_{j=0..n} j! * k^(j-1) * Stirling1(n,j) for n > 0.

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A006252 Expansion of e.g.f. 1/(1 - log(1+x)).

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A088501 Expansion of e.g.f. 1/(1-2*log(1+x)).

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A335531 Expansion of e.g.f. 1/(1-3*log(1+x)).

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A317172 a(n) = n! * [x^n] 1/(1 - n*log(1 + x)).

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A320079 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k*log(1 - x)).

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A354147 Expansion of e.g.f. 1/(1 - 4 * log(1+x)).

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A365604 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x)).

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A334369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - (k-1)*log(1 + x))/(1 - k*log(1 + x)).

A334369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - (k-1)log(1 + x))/(1 - klog(1 + x)).