A006252 -id:A006252 - cp's OEIS frontend

A351280 a(n) = Sum_{k=0..n} k! * k^k * Stirling1(n,k).

1, 1, 7, 140, 5254, 318854, 28455182, 3506576856, 570360248856, 118356589567440, 30512901324706608, 9566812017770347152, 3584662956711860108352, 1581905384865801328253712, 812047187127758913474118032, 479763784808095613489811245568

Offset: 0

Seiichi Manyama, Feb 06 2022

nonn

Cf. A006252, A305819, A320083, A350721, A350725, A351281.

a[0] = 1; a[n_] := Sum[k! * k^k * StirlingS1[n, k], {k, 1, n}]; Array[a, 16, 0] (* Amiram Eldar, Feb 06 2022 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*log(1+x))^k)))

E.g.f.: Sum_{k>=0} (k * log(1+x))^k.

a(n) ~ exp(-exp(-1)/2) * n! * n^n. - Vaclav Kotesovec, Feb 06 2022

A129841 Antidiagonal sums of triangle T defined in A048594: T(j,k) = k! * Stirling1(j,k), 1<= k <= j.

Original entry on oeis.org

1, -1, 4, -12, 52, -256, 1502, -10158, 78360, -680280, 6574872, -70075416, 816909816, -10342968456, 141357740736, -2074340369088, 32530886655168, -542971977209760, 9610316495698416, -179788450082431536, 3544714566466060032

Offset: 1

Paul Curtz, May 22 2007

sign

			First seven rows of T are
[    1 ]
[   -1,      2 ]
[    2,     -6,      6 ]
[   -6,     22,    -36,     24 ]
[   24,   -100,    210,   -240,    120 ]
[ -120,    548,  -1350,   2040,  -1800,    720 ]
[  720,  -3528,   9744, -17640,  21000, -15120,   5040 ]

P. Curtz, Integration numerique des systemes differentiels a conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, 1969, 135 pages, p. 61. Available from Centre d'Electronique de L'Armement, 35170 Bruz, France, or INRIA, Projets Algorithmes, 78150 Rocquencourt.
P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.
P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, no. 55, pp. 67-78.

Cf. A048594 (T read by rows), A075181 (T unsigned with rows read backwards), A006252 (row sums of T), A000142 (main diagonal of T), A001286 (unsigned first subdiagonal of T). Unsigned values of second through sixth column of T are in A052517, A052748, A052753, A052767, A052779 resp.

m:=21; T:=[ [ Factorial(k)*StirlingFirst(j, k): k in [1..j] ]: j in [1..m] ]; [ &+[ T[j-k+1][k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; // Klaus Brockhaus, Jun 03 2007

m = 21; t[j_, k_] := k!*StirlingS1[j, k]; Total /@ Table[ t[j-k+1, k], {j, 1, m}, {k, 1, Quotient[j+1, 2]}] (* Jean-François Alcover, Aug 13 2012, translated from Klaus Brockhaus's Magma program *)

E.g.f. for k-th column (k>=1): log(1+x)^k. For further formulas see the references.

Edited and extended by Klaus Brockhaus, Jun 03 2007

A305307 Expansion of e.g.f. 1/(1 - log(1 + x)/(1 - x)).

Original entry on oeis.org

1, 1, 3, 17, 120, 1084, 11642, 146446, 2101656, 33958344, 609431232, 12033015840, 259163792016, 6047213451408, 151953760489008, 4091057804809104, 117485988199385088, 3584814699783432960, 115816462543697120640, 3949619921174717629056, 141780511159572486530304, 5344008726418981985707776

Offset: 0

Ilya Gutkovskiy, May 29 2018

nonn

a(n)/n! is the invert transform of [1, 1 - 1/2, 1 - 1/2 + 1/3, 1 - 1/2 + 1/3 - 1/4, 1 - 1/2 + 1/3 - 1/4 + 1/5, ...].

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 120*x^4/4! + 1084*x^5/5! + 11642*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..410
N. J. A. Sloane, Transforms

Cf. A006252, A024167, A058312, A058313, A305306.

g:= proc(n) g(n):= `if`(n=1, 0, g(n-1))-(-1)^n/n end:
b:= proc(n) option remember; `if`(n=0, 1,
      add(g(j)*b(n-j), j=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20);  # Alois P. Heinz, May 29 2018

nmax = 21; CoefficientList[Series[1/(1 - Log[1 + x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/(1 - Sum[Sum[(-1)^(j + 1)/j, {j, 1, k}] x^k , {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[((-1)^(k + 1) LerchPhi[-1, 1, k + 1] + Log[2]) a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 21}]

E.g.f.: 1/(1 - Sum_{k>=1} (A058313(k)/A058312(k))*x^k).

a(n) ~ n! * (2 - LambertW(exp(2))) / ((1 + 1/LambertW(exp(2))) * (LambertW(exp(2)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021

A330150 Expansion of e.g.f. exp(-x) / (1 - log(1 + x)).

Original entry on oeis.org

1, 0, 0, 1, -1, 8, -16, 159, -659, 6824, -46680, 517581, -4941685, 61043344, -735256328, 10269016939, -147207286503, 2322683458544, -38298239486672, 677630804946393, -12581447014620585, 247342217288517496, -5096876494438056928, 110338442309322274295

Offset: 0

Ilya Gutkovskiy, Dec 03 2019

sign

Inverse binomial transform of A006252.

Cf. A052841, A330149.

Cf. A006252, A291981.

nmax = 23; CoefficientList[Series[Exp[-x]/(1 - Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * A006252(k).

a(n) = (-1)^n + Sum_{k=1..n} (-1)^(k-1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Dec 19 2023

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.

A347023 E.g.f.: 1 / (1 - 6 * log(1 + x))^(1/6).

Original entry on oeis.org

1, 1, 6, 72, 1254, 28794, 819888, 27869316, 1101032100, 49570797780, 2505156062472, 140417898936336, 8644973807845368, 579908437058338920, 42098286646367326368, 3288252917244250703664, 274974019392668843164176, 24510436934573885695407504, 2319947117871178825560902112

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

In general, for k > 1, if e.g.f. = 1 / (1 - k*log(1 + x))^(1/k), then a(n) ~ n! * exp(1/k^2) / (Gamma(1/k) * k^(1/k) * n^(1 - 1/k) * (exp(1/k) - 1)^(n + 1/k)). - Vaclav Kotesovec, Aug 14 2021

Cf. A006252, A008542, A320343, A346985, A347019, A347020, A347021, A347022.

nmax = 18; CoefficientList[Series[1/(1 - 6 Log[1 + x])^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling1(n,k) * A008542(k).

a(n) ~ n! * exp(1/36) / (Gamma(1/6) * 6^(1/6) * n^(5/6) * (exp(1/6) - 1)^(n + 1/6)). - Vaclav Kotesovec, Aug 14 2021

A354134 Expansion of e.g.f. 1/(1 - log(1 + x)^3/6).

Original entry on oeis.org

1, 0, 0, 1, -6, 35, -205, 1204, -6692, 29084, 17160, -3069924, 61356724, -959574408, 13499619224, -174983776176, 2029529618080, -18417948918640, 36189097244720, 4235753092128480, -157628320980720480, 4166967770825777280, -95152715945973322560

Offset: 0

Seiichi Manyama, May 18 2022

sign

Cf. A006252, A354135.

Cf. A346894, A346922.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x)^3/6)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 3, 1)*v[i-j+1])); v;

a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 1)/6^k);

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling1(k,3) * a(n-k).

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling1(n,3*k)/6^k.

A354135 Expansion of e.g.f. 1/(1 - log(1 + x)^5/120).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, -15, 175, -1960, 22449, -269073, 3403070, -45510630, 643152796, -9586136560, 150319669136, -2473024029840, 42562037379744, -764017130370276, 14260496108114340, -275877454002406236, 5512350021871343616, -113318466860425703184

Offset: 0

Seiichi Manyama, May 18 2022

sign

Cf. A006252, A354134.

Cf. A346920, A346924.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x)^5/120)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 5, 1)*v[i-j+1])); v;

a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 1)/120^k);

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling1(k,5) * a(n-k).

a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling1(n,5*k)/120^k.

A354229 Expansion of e.g.f. 1/(1 - log(1 + x)^3).

Original entry on oeis.org

1, 0, 0, 6, -36, 210, -630, -5376, 153048, -2194296, 22190760, -93956544, -2677330656, 97821857952, -2019503487456, 27899293618944, -98409183995520, -9548919666829440, 410311098024923520, -10652005874894469120, 176525303194482117120, -46197517147757867520

Offset: 0

Seiichi Manyama, May 20 2022

sign

Cf. A006252, A354230.

Cf. A353118, A354134, A354231.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x)^3)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i, j)*stirling(j, 3, 1)*v[i-j+1])); v;

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

a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * Stirling1(k,3) * a(n-k).

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling1(n,3*k).

A354237 Expansion of e.g.f. 1 / (1 - log(1 + 2*x) / 2).

Original entry on oeis.org

1, 1, 0, 2, -8, 64, -592, 6768, -90624, 1395840, -24292608, 471453696, -10094066688, 236340378624, -6007053852672, 164713554069504, -4846361933021184, 152300800682754048, -5091189648734748672, 180386551596145508352, -6752521487083688165376

Offset: 0

Ilya Gutkovskiy, Jun 06 2022

sign

Cf. A006252, A088500, A088501, A122704, A155585, A227917, A354750, A354751.

nmax = 20; CoefficientList[Series[1/(1 - Log[1 + 2 x]/2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 2^(n - k), {k, 0, n}], {n, 0, 20}]

my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+2*x)/2))) \\ Michel Marcus, Jun 06 2022

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * 2^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * (-2)^(k-1) * a(n-k).
a(n) ~ n! * (-1)^(n+1) * 2^(n+1) / (n * log(n)^2) * (1 - (4 + 2*gamma)/log(n) + (12 + 12*gamma + 3*gamma^2 - Pi^2/2)/log(n)^2 + (2*Pi^2*gamma - 32 + 4*Pi^2 - 24*gamma^2 - 8*zeta(3) - 4*gamma^3 - 48*gamma)/log(n)^3 + (80 - 20*Pi^2*gamma + 40*zeta(3)*gamma - 5*Pi^2*gamma^2 + 160*gamma + 5*gamma^4 + 80*zeta(3) + 40*gamma^3 + Pi^4/12 - 20*Pi^2 + 120*gamma^2)/log(n)^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 06 2022

cp's OEIS Frontend

A351280 a(n) = Sum_{k=0..n} k! * k^k * Stirling1(n,k).

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A129841 Antidiagonal sums of triangle T defined in A048594: T(j,k) = k! * Stirling1(j,k), 1<= k <= j.

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Magma

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A305307 Expansion of e.g.f. 1/(1 - log(1 + x)/(1 - x)).

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Maple

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A330150 Expansion of e.g.f. exp(-x) / (1 - 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)).

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A347023 E.g.f.: 1 / (1 - 6 * log(1 + x))^(1/6).

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A354134 Expansion of e.g.f. 1/(1 - log(1 + x)^3/6).

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A354135 Expansion of e.g.f. 1/(1 - log(1 + x)^5/120).

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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 - klog(1 + x)).