A002104 -id:A002104 - cp's OEIS frontend

A191365 Expansion of e.g.f. (1/(1-x))^exp(x).

1, 1, 4, 18, 102, 695, 5485, 49077, 490308, 5404569, 65106103, 850535477, 11972432846, 180605413001, 2906109200293, 49678357272247, 898988188301320, 17167497793440977, 344991795682802331, 7277230501449340417, 160765066207998479698

Offset: 0

Vladimir Kruchinin, May 31 2011

nonn

Exponential transform of A002104. - Seiichi Manyama, May 03 2022

Seiichi Manyama, Table of n, a(n) for n = 0..448

Cf. A002104, A298374.

CoefficientList[Series[(1/(1-x))^Exp[x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 21 2013 *)

a(n):=sum(sum(binomial(n,i)*k^i*(-1)^(n-k-i)*stirling1(n-i,k),i,0,n-k),k,1,n);

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x)^exp(x))) \\ Seiichi Manyama, May 03 2022

a(n) = sum(k=1..n, sum(i=0..n-k, binomial(n,i)*k^i*(-1)^(n-k-i)*Stirling1(n-i,k))), n>0, a(0)=1.
a(n) ~ n! * n^(exp(1)-1)/Gamma(exp(1)) * (1-exp(1)*(exp(1)-1)*log(n)/n). - Vaclav Kotesovec, Jun 21 2013
a(0) = 1; a(n) = Sum_{k=1..n} A002104(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, May 03 2022

A052863 Expansion of e.g.f. log(-1/(-1+x))*exp(x) - log(-1/(-1+x)).

Original entry on oeis.org

0, 0, 2, 6, 18, 65, 295, 1652, 11032, 85353, 749203, 7347384, 79564496, 942541041, 12121319327, 168145213732, 2502276609008, 39761200642225, 671855234838915, 12028625060491336, 227451564319791336, 4529507975800063337, 94751047516476943359, 2077192015403191663844

Offset: 0

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

Previous name was: A simple grammar.

Seiichi Manyama, Table of n, a(n) for n = 0..450
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 831

spec := [S,{B=Set(Z,1 <= card),C=Cycle(Z),S=Prod(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[Log[-1/(-1+x)]*E^x-Log[-1/(-1+x)], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 29 2013 *)

my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(-log(1-x)*(exp(x)-1)))) \\ Seiichi Manyama, May 13 2022

a(n) = sum(k=0, n-2, k!*binomial(n, k+1)); \\ Seiichi Manyama, May 13 2022

E.g.f.: log(-1/(-1+x))*exp(x) - log(-1/(-1+x)).
Recurrence: {a(1)=0, a(3)=6, a(2)=2, (-n^3-2*n-3*n^2)*a(n)+(19*n+11*n^2+2*n^3+10)*a(n+1)+(-38*n-12*n^2-n^3-36)*a(n+2)+(41+26*n+4*n^2)*a(n+3)+(-17-5*n)*a(n+4)+2*a(n+5), a(4)=18, a(5)=65}
a(n) = A002104(n)-(n-1)!. - Vladeta Jovovic, Apr 03 2005
a(n) ~ (n-1)! * (exp(1)-1). - Vaclav Kotesovec, Sep 29 2013
a(n) = Sum_{k=0..n-2} k! * binomial(n,k+1). - Seiichi Manyama, May 13 2022

New name using e.g.f., Joerg Arndt, Sep 30 2013

A177699 Expansion of e.g.f. log(1+x) * sinh(x).

Original entry on oeis.org

0, 0, 2, -3, 12, -40, 190, -1071, 7224, -56232, 495898, -4880755, 53005700, -629398848, 8110146070, -112690225935, 1679413757168, -26719024870576, 451969255722162, -8099650628337987, 153288815339260796, -3054957193416951480, 63949589015139119598, -1402819397613793354063

Offset: 0

Michel Lagneau, May 11 2010

sign

			log(1+x) * sinh(x) = x^2 -x^3/2 +x^4/2 -x^5/3 +19*x^6/72 -17*x^7/80 +...

L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.

Cf. A002104, A009410, A009416, A381016.

A177699 := proc(n)
        log(1+x)*sinh(x) ;
        coeftayl(%,x=0,n)*n! ;
end proc;
seq(A177699(n),n=0..20) ; # R. J. Mathar, Nov 07 2011

Table[n ((-1)^n HypergeometricPFQ[{1, 1, 1 - n}, {2}, -1] + HypergeometricPFQ[{1, 1, 1 - n}, {2}, 1])/2, {n, 1, 20}] (* Benedict W. J. Irwin, May 30 2016 *)

a(n) = (-1)^n*sum(k=1, n\2, (n-2*k)!*binomial(n, 2*k-1)); \\ Seiichi Manyama, Feb 12 2025

a(n) = n*((-1)^n*3F1(1,1,1-n;2;-1)+3F1(1,1,1-n;2;1))/2, n>0. - Benedict W. J. Irwin, May 30 2016
a(n) ~ (-1)^n * (n-1)! * sinh(1). - Vaclav Kotesovec, May 30 2016
a(n) = (-1)^n * Sum_{k=1..floor(n/2)} (n-2*k)! * binomial(n,2*k-1). - Seiichi Manyama, Feb 12 2025

A346395 Expansion of e.g.f. -log(1 - x) * exp(3*x).

Original entry on oeis.org

0, 1, 7, 38, 192, 969, 5115, 29322, 187992, 1370745, 11392839, 107043606, 1122823944, 12989320785, 164040593067, 2243143392138, 32994768719376, 519229765892241, 8701862242296807, 154700700117472422, 2907409255935736752, 57588370882960384377, 1198954118077558162875

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A002104, A053486, A346394, A346396.

nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[3 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[3^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 22}]

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i+2)*v[i]-3*(i-1)*v[i-1]+3^(i-1)); v; \\ Seiichi Manyama, May 27 2022

a(n) = n! * Sum_{k=0..n-1} 3^k / ((n-k) * k!).
a(n) ~ exp(3) * (n-1)!. - Vaclav Kotesovec, Aug 09 2021
a(0) = 0, a(1) = 1, a(n) = (n+2) * a(n-1) - 3 * (n-1) * a(n-2) + 3^(n-1). - Seiichi Manyama, May 27 2022

A001338 -1 + Sum (k-1)! C(n,k), k = 1..n for n > 0, a(0) = 1.

Original entry on oeis.org

1, 0, 2, 7, 23, 88, 414, 2371, 16071, 125672, 1112082, 10976183, 119481295, 1421542640, 18348340126, 255323504931, 3809950977007, 60683990530224, 1027542662934914, 18430998766219335, 349096664728623335, 6962409983976703336, 145841989688186383358

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
E. Biondi, L. Divieti, G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math. 22 1970 22-35.

Partial sums of A000522.

Equals A002104(n) + 1.

Join[{1}, Table[-1 + Sum[(k - 1)! Binomial[n, k], {k, n}], {n, 20}]] (* T. D. Noe, Jun 28 2012 *)

Conjecture: a(n) +(-n-1)*a(n-1) +2*(n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Feb 16 2014

a(n) = n*a(n-1) - (n-1)*a(n-2) - 1, with a sign reversal for n>=2. - Richard R. Forberg, Dec 16 2014

A260323 Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=-1.

Original entry on oeis.org

1, 3, 2, 8, 6, 6, 24, 24, 24, 24, 89, 80, 60, 120, 120, 415, 450, 480, 360, 720, 720, 2372, 2142, 2730, 840, 2520, 5040, 5040, 16072, 17696, 10416, 21840, 6720, 20160, 40320, 40320, 125673, 112464, 151704, 184464, 15120, 60480, 181440, 362880, 362880

Offset: 1

N. J. A. Sloane, Jul 23 2015

			Triangle begins:
1,
3,2,
8,6,6,
24,24,24,24,
89,80,60,120,120,
415,450,480,360,720,720,
2372,2142,2730,840,2520,5040,5040,
...

J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. Gives first 10 rows. [Annotated scanned copy]

Rows, column sums give A002104, A002742, A002745, A002746.

Cf. A260322-A260325.

A260323 := proc(n,r)
    if r = 0 then
        1 ;
    elif n > r+1 then
        0 ;
    else
        add( 1/(r-j*n)!/j,j=1..(r)/n) ;
        %*r! ;
    end if;
end proc:
for r from 1 to 20 do
    for n from 1 to r do
        printf("%a,",A260323(n,r)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 24 2015

T[n_, k_] := If[n == 0, 1, If[k > n+1, 0, Sum[1/(n - j*k)!/j, {j, 1, n/k}]]]*n!;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 25 2023, after R. J. Mathar *)

A346394 Expansion of e.g.f. -log(1 - x) * exp(2*x).

Original entry on oeis.org

0, 1, 5, 20, 78, 324, 1520, 8336, 53872, 405600, 3492416, 33798016, 362543104, 4264455168, 54540715008, 753246711808, 11168972683264, 176937613586432, 2982069587042304, 53271637651996672, 1005385746384846848, 19987620914387812352, 417489079682758213632

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A002104, A010842, A066534, A346395, A346396.

nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[2 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[2^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 22}]

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i+1)*v[i]-2*(i-1)*v[i-1]+2^(i-1)); v; \\ Seiichi Manyama, May 27 2022

a(n) = n! * Sum_{k=0..n-1} 2^k / ((n-k) * k!).
a(n) = Sum_{k=0..n} binomial(n,k) * A002104(k).
a(n) ~ exp(2) * (n-1)!. - Vaclav Kotesovec, Aug 09 2021
a(0) = 0, a(1) = 1, a(n) = (n+1) * a(n-1) - 2 * (n-1) * a(n-2) + 2^(n-1). - Seiichi Manyama, May 27 2022

A346396 Expansion of e.g.f. -log(1 - x) * exp(4*x).

Original entry on oeis.org

0, 1, 9, 62, 390, 2384, 14680, 93680, 635824, 4697664, 38442112, 351331584, 3582715136, 40476303360, 501863078912, 6767130867712, 98464775493632, 1536203429306368, 25564684461735936, 451816479967608832, 8448863295040978944, 166627401783086415872, 3455980532191764676608

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A002104, A053487, A346394, A346395.

nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[4 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[4^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 22}]

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i+3)*v[i]-4*(i-1)*v[i-1]+4^(i-1)); v; \\ Seiichi Manyama, May 27 2022

a(n) = n! * Sum_{k=0..n-1} 4^k / ((n-k) * k!).
a(n) ~ exp(4) * (n-1)!. - Vaclav Kotesovec, Aug 09 2021
a(0) = 0, a(1) = 1, a(n) = (n+3) * a(n-1) - 4 * (n-1) * a(n-2) + 4^(n-1). - Seiichi Manyama, May 27 2022

A353546 Expansion of e.g.f. -log(1-2x) exp(x)/2.

Original entry on oeis.org

0, 1, 4, 17, 96, 729, 7060, 83033, 1146656, 18164625, 324488068, 6450956929, 141233271872, 3376008830505, 87480173354964, 2442396780039817, 73089894980585408, 2333809837398044321, 79198287879591647364, 2846319497398561356913

Offset: 0

Seiichi Manyama, May 27 2022

nonn

Cf. A002104, A353547, A353548, A353549.
Cf. A346394.
Essentially partial sums of A010844.

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

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

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(2*i-1)*v[i]-2*(i-1)*v[i-1]+1); v;

a(n) = n! * Sum_{k=0..n-1} 2^(n-1-k) / ((n-k) * k!).
a(0) = 0, a(1) = 1, a(n) = (2 * n - 1) * a(n-1) - 2 * (n-1) * a(n-2) + 1.
a(n) ~ (n-1)! * exp(1/2) * 2^(n-1). - Vaclav Kotesovec, Jun 08 2022

A353547 Expansion of e.g.f. -log(1-3x) exp(x)/3.

Original entry on oeis.org

0, 1, 5, 30, 256, 2969, 43665, 776194, 16159304, 385353945, 10353609253, 309401268494, 10177974023448, 365446593201793, 14220922741157249, 596150920955286402, 26783000840591098288, 1283751796983110068817, 65389160400251577565797

Offset: 0

Seiichi Manyama, May 27 2022

nonn

Cf. A002104, A353546, A353548, A353549.
Cf. A346395.
Essentially partial sums of A010845.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-log(1-3*x)*exp(x)/3)))

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

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(3*i-2)*v[i]-3*(i-1)*v[i-1]+1); v;

a(n) = n! * Sum_{k=0..n-1} 3^(n-1-k) / ((n-k) * k!).
a(0) = 0, a(1) = 1, a(n) = (3 * n - 2) * a(n-1) - 3 * (n-1) * a(n-2) + 1.
a(n) ~ (n-1)! * exp(1/3) * 3^(n-1). - Vaclav Kotesovec, Jun 08 2022

cp's OEIS Frontend

A191365 Expansion of e.g.f. (1/(1-x))^exp(x).

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A052863 Expansion of e.g.f. log(-1/(-1+x))*exp(x) - log(-1/(-1+x)).

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A177699 Expansion of e.g.f. log(1+x) * sinh(x).

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A346395 Expansion of e.g.f. -log(1 - x) * exp(3*x).

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A001338 -1 + Sum (k-1)! C(n,k), k = 1..n for n > 0, a(0) = 1.

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A260323 Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=-1.

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A346394 Expansion of e.g.f. -log(1 - x) * exp(2*x).

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A346396 Expansion of e.g.f. -log(1 - x) * exp(4*x).

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A353546 Expansion of e.g.f. -log(1-2x) exp(x)/2.

A353547 Expansion of e.g.f. -log(1-3x) exp(x)/3.