A007840 -id:A007840 - cp's OEIS frontend

A302548 Expansion of e.g.f. -log(1 + log(1 - x))/(1 + log(1 - x)).

0, 1, 4, 22, 155, 1333, 13541, 158688, 2107682, 31291894, 513590170, 9234669420, 180534475832, 3812852144788, 86517295628188, 2099170738243328, 54233876338638192, 1486517654443664016, 43084555863325589232, 1316588795487600071904, 42306543064537291007424, 1426115146736949130634400

Offset: 0

Ilya Gutkovskiy, Jun 20 2018

nonn

			E.g.f.: A(x) = x + 4*x^2/2! + 22*x^3/3! + 155*x^4/4! + 1333*x^5/5! + 13541*x^6/6! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..418

Cf. A000254, A001008, A002805, A003713, A007840, A073596, A222058, A300490, A302547.

H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*H(k)*k!, k=1..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 21 2018

nmax = 21; CoefficientList[Series[-Log[1 + Log[1 - x]]/(1 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] HarmonicNumber[k] k!, {k, 0, n}], {n, 0, 21}]

a(n) = Sum_{k=1..n} |Stirling1(n,k)|*H(k)*k!, where H(k) is the k-th harmonic number.

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

A302914 Determinant of n X n matrix whose main diagonal consists of the first n 10-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 9, 234, 11934, 1002456, 125307000, 21803418000, 5036589558000, 1490830509168000, 550116457882992000, 247552406047346400000, 133430746859519709600000, 84861955002654535305600000, 62882708656967010661449600000, 53701833193049827104877958400000

Offset: 1

Muniru A Asiru, Apr 15 2018

nonn

From Vaclav Kotesovec, Apr 16 2018: (Start)
In general, for k > 2, these determinants for k-gonal numbers satisfies:
a(n,k) = ((k-2)/2)^(n-1) * Gamma(n) * Gamma(n + k/(k-2)) / Gamma(1 + k/(k-2)).
a(n,k) ~ 4*Pi * (k/2 - 1)^n * n^(2*n + 2/(k-2)) / (k * Gamma(k/(k-2)) * exp(2*n)).
a(n+1,k) = a(n,k) * n*((k-2)*n + k)/2.
(End)

			The matrix begins:
  1   1   1   1   1   1   1 ...
  1  10   1   1   1   1   1 ...
  1   1  27   1   1   1   1 ...
  1   1   1  52   1   1   1 ...
  1   1   1   1  85   1   1 ...
  1   1   1   1   1 126   1 ...
  1   1   1   1   1   1 175 ...

Cf. A001107.
Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), A302910 (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), this sequence (k=10).
Cf. A007840 (permanent instead of determinant, for k=2).

d:=(i,j)->`if`(i<>j,1,i*(4*i-3)):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);

nmax = 20; Table[Det[Table[If[i == j, i*(4*i-3), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
RecurrenceTable[{a[n+1] == a[n] * n*(4*n + 5), a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[4^(n+1) * Gamma[n] * Gamma[n + 5/4] / (5*Gamma[1/4])], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(4*i-3)))); \\ Michel Marcus, Apr 16 2018

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = 4^(n+1) * Gamma(n) * Gamma(n + 5/4) / (5*Gamma(1/4)).
a(n) ~ Pi * 2^(2*n + 3) * n^(2*n + 1/4) / (5 * Gamma(1/4) * exp(2*n)).
a(n+1) = a(n) * n*(4*n + 5).
(End)

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

A320502 a(n) = Sum_{k=0..n} (k!)^2 * abs(Stirling1(n,k)).

Original entry on oeis.org

1, 1, 5, 50, 842, 21644, 792676, 39297600, 2536525008, 206794669104, 20785423425264, 2525457805492896, 364910211591903072, 61847041340997089280, 12151693924459271926272, 2739901558132307387349504, 702704348810821821056454144, 203409730893592265642619623424

Offset: 0

Vaclav Kotesovec, Oct 13 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..253

Cf. A000670, A007840, A064618, A192554.

[(&+[Abs(StirlingFirst(n,k))*(Factorial(k))^2: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Oct 14 2018

Table[Sum[Abs[StirlingS1[n, k]]*k!^2, {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, k!^2*abs(stirling(n, k, 1))); \\ Michel Marcus, Oct 14 2018

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, k!*(-log(1-x))^k))) \\ Seiichi Manyama, Apr 22 2022

a(n) ~ exp(1/2) * (n!)^2.

E.g.f.: Sum_{k>=0} k! * (-log(1-x))^k. - Seiichi Manyama, Apr 22 2022

A336258 a(0) = 1; a(n) = (n!)^2 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^2.

Original entry on oeis.org

1, 1, 5, 58, 1208, 39476, 1861372, 119587224, 10040970816, 1067383279872, 140110136642304, 22256626639796352, 4207858001708629248, 933704296260740939520, 240293228328619963492608, 70992050129486593239246336, 23863916105454465092261412864

Offset: 0

Ilya Gutkovskiy, Jul 15 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..247

Cf. A007840, A074707, A102221, A336243, A336259, A336260, A336261.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)/i^2, i=1..n))
    end:
a:= n-> n!^2*b(n):
seq(a(n), n=0..16);  # Alois P. Heinz, Jan 04 2024

a[0] = 1; a[n_] := a[n] = (n!)^2 Sum[a[k]/(k! (n - k))^2, {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[1/(1 - PolyLog[2, x]), {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = (n!)^2 * [x^n] 1 / (1 - polylog(2,x)).

a(n) ~ (n!)^2 / (-log(1-r) * r^n), where r = 0.76154294453204558806805187241... is the root of the equation polylog(2,r) = 1. - Vaclav Kotesovec, Jul 15 2020

A351136 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(2n) Stirling1(n,k).

Original entry on oeis.org

1, 1, 33, 4760, 1814698, 1436035954, 2041681617638, 4736066140912728, 16729538152432476024, 85437808930634601070944, 605822464949212598847700512, 5774077466357788471179323050704, 72030066703292325305595937373723040

Offset: 0

Seiichi Manyama, Feb 02 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..160

Cf. A007840, A320096, A351137.

Cf. A187755, A351133, A351138.

a[0] = 1; a[n_] := Sum[(-1)^(n - k) * k! * k^(2*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 13, 0] (* Amiram Eldar, Feb 02 2022 *)

a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^(2*n)*stirling(n, k, 1));

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k^2*x))^k)))

E.g.f.: Sum_{k>=0} (-log(1 - k^2*x))^k.

a(n) ~ c * r^(2*n) * (1 + r*exp(2/r))^n * n^(3*n + 1/2) / exp(3*n), where r = 0.9414380538633895499299457441124149470954491698433... is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-2/r) and c = 2.22047212763474863127102273073825610210704559048894... - Vaclav Kotesovec, Feb 03 2022

A352069 Expansion of e.g.f. 1 / (1 + log(1 - 3*x) / 3).

Original entry on oeis.org

1, 1, 5, 42, 492, 7374, 134478, 2887128, 71281656, 1988802720, 61860849552, 2121993490176, 79566300371952, 3237181141173264, 142019158472311248, 6682603650677875584, 335698708873243355136, 17930674324049810882688, 1014685181110897126616448, 60641642160287342580586752

Offset: 0

Ilya Gutkovskiy, Mar 02 2022

nonn

Cf. A007840, A032031, A087674, A227917, A255927, A352071.

nmax = 19; CoefficientList[Series[1/(1 + Log[1 - 3 x]/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! (-3)^(n - k), {k, 0, n}], {n, 0, 19}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-3*x)/3))) \\ Michel Marcus, Mar 02 2022

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-3)^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * 3^(k-1) * a(n-k).
a(n) ~ n! * 3^(n+1) * exp(3*n) / (exp(3) - 1)^(n+1). - Vaclav Kotesovec, Mar 03 2022

A354123 Expansion of e.g.f. 1/(1 + log(1 - x))^4.

Original entry on oeis.org

1, 4, 24, 188, 1804, 20416, 265640, 3901320, 63776280, 1147796160, 22540858080, 479500074720, 10980929163360, 269298981833280, 7040446188020160, 195439047629422080, 5740498087530831360, 177855276360034736640, 5796391124741936993280

Offset: 0

Seiichi Manyama, May 17 2022

nonn

Cf. A007840, A052801, A354122.

Cf. A226738, A354121.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x))^4))

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

a(n) = (1/6) * Sum_{k=0..n} (k + 3)! * |Stirling1(n,k)|.
a(n) ~ sqrt(Pi/2) * n^(n + 7/2) / (3 * (exp(1) - 1)^(n+4)). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (3*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A355086 E.g.f. A(x) satisfies A(x) = 1 - log(1-x) * A(2*x).

Original entry on oeis.org

1, 1, 5, 68, 2318, 191364, 37322176, 16851654336, 17323677619888, 39991811695203552, 204958165376127918144, 2309776412016044230960128, 56778926016923229432156258048, 3023733345610004146919028796718592

Offset: 0

Seiichi Manyama, Jun 18 2022

sign

Cf. A007840, A355087.

Cf. A352860.

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

a(0) = 1; a(n) = Sum_{k=1..n} 2^(n-k) * (k-1)! * binomial(n,k) * a(n-k).

A052860 A simple grammar: rooted sequences of cycles.

Original entry on oeis.org

0, 1, 2, 9, 56, 440, 4164, 46046, 582336, 8288136, 131090880, 2280970032, 43298796672, 890441326320, 19720847692896, 467964024901200, 11844861486802944, 318549937907204352, 9070876711252816128, 272648086802525651328, 8626452694650322744320

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Note that here the root is not allowed to be part of the sequence of cycles. We select a root and then form sequences from the cycles in the permutations of the remaining n-1 elements. Cf. A218817. - Geoffrey Critzer, Nov 06 2012

Alois P. Heinz, Table of n, a(n) for n = 0..150
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 828

Cf. A007840.

spec := [S,{C=Cycle(Z),B=Sequence(C),S=Prod(Z,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

nn=20;a=Log[1/(1-x)];Range[0,nn]!CoefficientList[Series[x/(1-a) ,{x,0,nn}],x]  (* Geoffrey Critzer, Nov 06 2012 *)

a(n)=n!*polcoeff(x/(1+log(1-x +x*O(x^n))),n) \\ Paul D. Hanna, Jul 19 2006

E.g.f.: -1/(-1+log(-1/(-1+x)))*x.
a(n) = n*A007840(n-1). a(n) = n!*Sum_{k=0..n-1} a(k)/k!/(n-k) for n>=1 with a(0)=0. - Paul D. Hanna, Jul 19 2006
a(n) ~ n! * exp(n-1) / (exp(1)-1)^n. - Vaclav Kotesovec, Mar 16 2014

cp's OEIS Frontend

A302548 Expansion of e.g.f. -log(1 + log(1 - x))/(1 + log(1 - x)).

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A302914 Determinant of n X n matrix whose main diagonal consists of the first n 10-gonal numbers and all other elements are 1's.

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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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A320502 a(n) = Sum_{k=0..n} (k!)^2 * abs(Stirling1(n,k)).

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A336258 a(0) = 1; a(n) = (n!)^2 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^2.

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A351136 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(2*n) * Stirling1(n,k).

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A352069 Expansion of e.g.f. 1 / (1 + log(1 - 3*x) / 3).

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A354123 Expansion of e.g.f. 1/(1 + log(1 - x))^4.

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A351136 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(2n) Stirling1(n,k).