A000629 -id:A000629 - cp's OEIS frontend

A346954 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^4 / 4! ).

1, 10, 65, 350, 1736, 9030, 60355, 561550, 6188996, 69919850, 781211795, 8854058850, 106994019406, 1433756147470, 21287253921635, 339206526695750, 5630710652048216, 96341917117951890, 1708973354556320875, 31787279786739738250, 623964823224788294426

Offset: 4

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000453, A000629, A003704, A327505, A346390, A346895, A346955.

nmax = 24; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS2[n, 4] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 24}]

a(n) = Stirling2(n,4) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,4) * k * a(k).
a(n) ~ (n-1)! / (log(2^(3/4)*3^(1/4) + 1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/4)} (4*k)! * Stirling2(n,4*k)/(k * 24^k). - Seiichi Manyama, Jan 23 2025

A346955 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^5 / 5! ).

Original entry on oeis.org

1, 15, 140, 1050, 6951, 42651, 253660, 1594230, 12463451, 134921787, 1806513072, 25539589530, 355175465191, 4797717669123, 63797550625676, 860468790181686, 12275324511112971, 192498455326842819, 3353266112959628272, 63379650000684213834

Offset: 5

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000481, A000629, A003704, A327506, A346390, A346920, A346954.

nmax = 24; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^5/5!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
a[n_] := a[n] = StirlingS2[n, 5] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 5] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 5, 24}]

a(n) = Stirling2(n,5) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,5) * k * a(k).
a(n) ~ (n-1)! / (log(120^(1/5) + 1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/5)} (5*k)! * Stirling2(n,5*k)/(k * 120^k). - Seiichi Manyama, Jan 23 2025

A348468 Expansion of e.g.f. sqrt(exp(x)*(2-exp(x))).

Original entry on oeis.org

1, 0, -1, -3, -10, -45, -271, -2058, -18775, -199335, -2410516, -32683563, -490870315, -8087188200, -144994236661, -2810079139143, -58536519252130, -1304198088413265, -30946462816602331, -779104979758256298, -20742005411397108595, -582214473250983046155, -17184302765073000634276

Offset: 0

Michel Marcus, Oct 19 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425
Feng Qi and Mark Daniel Ward, Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function, arXiv:2110.08576 [math.CO], 2021.

Cf. A000629, A000918, A014304, A014307, A260701, A260902.

m = 22; Range[0, m]! * CoefficientList[Series[Sqrt[Exp[x]*(2 - Exp[x])], {x, 0, m}], x] (* Amiram Eldar, Oct 19 2021 *)

my(x='x+O('x^25)); Vec(serlaplace(sqrt(exp(x)*(2-exp(x)))))

a(n) ~ -sqrt(2) * n^(n-1) / (log(2)^(n - 1/2) * exp(n)). - Vaclav Kotesovec, Oct 21 2021

A355218 a(n) = Sum_{k>=1} (3*k - 1)^n / 2^k.

Original entry on oeis.org

1, 5, 43, 557, 9643, 208685, 5419243, 164184557, 5684837803, 221440158125, 9584118542443, 456289689634157, 23698327407870763, 1333388917719691565, 80794290325166308843, 5245268489291712773357, 363231496206350038884523, 26725646191850556128889005, 2082075690178933613292014443

Offset: 0

Ilya Gutkovskiy, Jun 24 2022

nonn

Cf. A000629, A000670, A007047, A080253, A151919, A328182, A355219, A355220.

nmax = 18; CoefficientList[Series[Exp[2 x]/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2^n + Sum[Binomial[n, k] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

E.g.f.: exp(2*x) / (2 - exp(3*x)).
a(0) = 1; a(n) = 2^n + Sum_{k=1..n} binomial(n,k) * 3^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * 3^k * A000670(k).
a(n) ~ n! * 3^n / (2^(1/3) * log(2)^(n+1)). - Vaclav Kotesovec, Jun 24 2022

A355219 a(n) = Sum_{k>=1} (4*k - 2)^n / 2^k.

Original entry on oeis.org

1, 6, 68, 1176, 27152, 783456, 27126848, 1095801216, 50589024512, 2627443262976, 151623974601728, 9624874873952256, 666516443992297472, 50002158357801885696, 4039720490206565777408, 349685083067909962039296, 32287291853754803207340032, 3167488677197974581176303616

Offset: 0

Ilya Gutkovskiy, Jun 24 2022

nonn

Cf. A000629, A000670, A007047, A080253, A285067, A328183, A355218, A355220.

nmax = 17; CoefficientList[Series[Exp[2 x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2^n + Sum[Binomial[n, k] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

E.g.f.: exp(2*x) / (2 - exp(4*x)).
a(0) = 1; a(n) = 2^n + Sum_{k=1..n} binomial(n,k) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n+k) * A000670(k).
a(n) ~ n! * 2^(2*n - 1/2) / log(2)^(n+1). - Vaclav Kotesovec, Jun 24 2022

A355220 a(n) = Sum_{k>=1} (4*k - 1)^n / 2^k.

Original entry on oeis.org

1, 7, 81, 1399, 32289, 931687, 32259441, 1303134679, 60160827969, 3124574220487, 180312309395601, 11445969681199159, 792626097462398049, 59462922484586318887, 4804064349575887075761, 415847988794676360818839, 38396277196654611908582529, 3766800071614388562865514887

Offset: 0

Ilya Gutkovskiy, Jun 24 2022

nonn

Cf. A000629, A000670, A080253, A259533, A285067, A328183, A355218, A355219.

nmax = 17; CoefficientList[Series[Exp[3 x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3^n + Sum[Binomial[n, k] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

E.g.f.: exp(3*x) / (2 - exp(4*x)).
a(0) = 1; a(n) = 3^n + Sum_{k=1..n} binomial(n,k) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * 4^k * A000670(k).
a(n) ~ n! * 2^(2*n - 1/4) / log(2)^(n+1). - Vaclav Kotesovec, Jun 24 2022

A052825 A simple grammar: partial sums of A008965.

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 18, 31, 50, 85, 144, 251, 438, 789, 1420, 2601, 4792, 8907, 16618, 31219, 58814, 111301, 211180, 401925, 766648, 1465899, 2808082, 5389509, 10360576, 19948155, 38460946, 74253513, 143527180, 277746975, 538048150, 1043342277, 2025049108

Offset: 0

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

Danny Rorabaugh, Table of n, a(n) for n = 0..2500
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 790

spec := [S,{B=Cycle(C),C=Sequence(Z,1 <= card),S=Prod(C,B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
h := n -> add(numtheory:-phi(j)/j*log((x^j-1)/(2*x^j-1)), j=1..n):
seq(coeff(series((x/(1-x))*h(n),x,n+1),x,n),n=0..36); # Peter Luschny, Oct 25 2015

m = 40;
gf = (x/(1-x))*Sum[EulerPhi[j]/j*Log[(x^j-1)/(2*x^j-1)], {j,1,m}] + O[x]^m;
CoefficientList[gf, x] (* Jean-François Alcover, Jun 03 2019 *)

var('x'); a = lambda n: taylor(x/(1-x) * sum([taylor(euler_phi(i)/i * log((x^i - 1)/(2*x^i - 1)), x, 0, n) for i in range(1, n+1)]), x, 0, n).coefficient(x^n) # Danny Rorabaugh, Oct 25 2015

G.f.: (x/(x-1))*Sum_{j>=1} (A000010(j)/j)*log((x^j-1)/(2*x^j-1)).

a(n) ~ 2^n/n * (1 + 2/n + 6/n^2 + 26/n^3 + 150/n^4 + 1082/n^5 + 9366/n^6 + 94586/n^7 + 1091670/n^8 + 14174522/n^9 + 204495126/n^10 + ...), for coefficients see A000629. - Vaclav Kotesovec, Jun 03 2019

More terms from Danny Rorabaugh, Oct 25 2015

A052861 E.g.f.: log((1-x)/(1-2x))x/(1-x).

Original entry on oeis.org

0, 0, 2, 15, 116, 1030, 10644, 127428, 1750944, 27325296, 479288160, 9355658400, 201405744000, 4743245520000, 121334466758400, 3350276227872000, 99309556729958400, 3145135939426252800

Offset: 0

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

Previous name was: A simple grammar.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 829

Cf. A000629.

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

CoefficientList[Series[Log[(1-x)/(1-2*x)]*x/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)
Flatten[{0,Table[FullSimplify[-n!*(EulerGamma + I*Pi + 2^n*LerchPhi[2,1,n] + PolyGamma[0,n])],{n,1,20}]}] (* Vaclav Kotesovec, Oct 01 2013 *)

E.g.f.: -log((-1+x)/(-1+2*x))*x/(-1+x).
Recurrence: {a(1)=0, a(0)=0, a(2)=2, (-2*n^4-12*n^3-22*n^2-12*n)*a(n)+(5*n^3+28*n^2+45*n+18)*a(n+1)+(-17*n-15-4*n^2)*a(n+2)+(n+2)*a(n+3)}.
a(n) ~ (n-1)! * 2^n * (1 + 2/n + 6/n^2 + 26/n^3 + 150/n^4 + 1082/n^5 + 9366/n^6 + 94586/n^7), coefficients are A000629. - Vaclav Kotesovec, Mar 17 2015

New name using e.g.f., Vaclav Kotesovec, Oct 01 2013

A080163 Sum of an infinite series: a(n) = Sum_{k>=0} ((k+1)(k+2))^n/(16(2^k)).

Original entry on oeis.org

1, 26, 1636, 191336, 35909776, 9877824416, 3744949458496, 1871860519454336, 1192747133878118656, 943718459840134969856, 907745644208033315808256, 1043182479702092427281524736, 1411605714773024334343061671936

Offset: 1

Karol A. Penson, Jan 31 2003

nonn

Table[1/8*Sum[Binomial[n,i]*(n+i)!*SeriesCoefficient[Exp[x]/(2-Exp[x]),{x,0,n+i}],{i,0,n}],{n,1,20}] (* Vaclav Kotesovec after Benoit Cloitre, Jun 29 2013 *)

a(n) = (1/8)*Sum_{i=0..n} C(n, i)*A000629(n+i). - Benoit Cloitre, Feb 02 2003

a(n) ~ (2n)!/(4*sqrt(2)*(log(2))^(2*n+1)). - Vaclav Kotesovec, Jun 29 2013

A090665 Triangle read by rows: T(n,k) = number of preferential arrangements of n things where the first object has rank k.

Original entry on oeis.org

1, 2, 1, 6, 5, 2, 26, 25, 18, 6, 150, 149, 134, 84, 24, 1082, 1081, 1050, 870, 480, 120, 9366, 9365, 9302, 8700, 6600, 3240, 720, 94586, 94585, 94458, 92526, 82320, 57120, 25200, 5040, 1091670, 1091669, 1091414, 1085364, 1038744, 871920, 554400, 221760, 40320

Offset: 1

Eugene McDonnell (eemcd(AT)mac.com), Dec 16 2003

The rows are the reverses of the rows of A054255.
Row sums give A000670.
Column 1 is A000629. - Joerg Arndt, Dec 08 2014
From Vincent Jackson, May 01 2023: (Start)
The formula
  T(n, k) = Sum_{i=k..n-1} i!*StirlingS2(n-1, i) + (k-1)!*StirlingS2(n-1,k-1)
can be derived by splitting the weak orders with the first object at rank k into three categories:
1. weak orders where another object (of the n-1 other objects) has rank k,
2. weak orders where all other objects have rank strictly less than k, and
3. weak orders where no other object is at rank k, but some object has rank greater than k.
The number of weak orders in the first category is Sum_{i=k..n-1} i!*StirlingS2(n-1, i), the number of weak orders of length n-1 with number of ranks between k and n-1 (i.e. A084416(n-1,k)). Given a weak order of length n-1 and number of ranks i >= k, the corresponding weak order of length n with the specified object at rank k is formed by inserting the new object into the appropriate rank.
The number of weak orders in the second category is (k-1)!*StirlingS2(n-1,k-1), the number of weak orders of length n-1 with number of ranks k-1. Given a weak order of length n-1 and number of ranks k-1, the corresponding weak order is formed by appending the new object in its own rank.
Lastly, the number of weak orders in the third category is (again) Sum_{i=k..n-1} i!*StirlingS2(n-1, i). Given a weak order of length n-1 and number of ranks k-1, the corresponding weak order is formed by inserting the new object in its own rank after the rank k-1, thereby shifting by one the ranks originally greater than or equal to k. (End)

			Triangle starts:
01: 1;
02: 2, 1;
03: 6, 5, 2;
04: 26, 25, 18, 6;
05: 150, 149, 134, 84, 24;
06: 1082, 1081, 1050, 870, 480, 120;
07: 9366, 9365, 9302, 8700, 6600, 3240, 720;
08: 94586, 94585, 94458, 92526, 82320, 57120, 25200, 5040;
09: 1091670, 1091669, 1091414, 1085364, 1038744, 871920, 554400, 221760, 40320;
10: 14174522, 14174521, 14174010, 14155350, 13950720, 12930120, 10190880, 5957280, 2177280, 362880;
...

Cf. A054255, A008277, A028246, A083410.

T = {n, k} |-> 2*Sum[i!*StirlingS2[n-1, i], {i, k, n-1}] + (k-1)i!*StirlingS2[n-1, k-1]  (* Vincent Jackson, May 01 2023 *)

From Vincent Jackson, May 01 2023: (Start)
T(n, k) = 2*(Sum_{i=k..n-1} i!*StirlingS2(n-1, i)) + (k-1)!*StirlingS2(n-1,k-1).
T(n, k) = 2*A084416(n-1,k) + (k-1)!*StirlingS2(n-1,k-1).
T(n, k) = A084416(n-1,k) + A084416(n-1,k-1). (End)
Sum_{k=1..n} k * T(n,k) = A083410(n). - Alois P. Heinz, Feb 20 2025

Corrected by Alois P. Heinz, Dec 08 2014

Name clarified by Vincent Jackson, May 01 2023

cp's OEIS Frontend

A346954 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^4 / 4! ).

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A346955 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^5 / 5! ).

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A348468 Expansion of e.g.f. sqrt(exp(x)*(2-exp(x))).

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A355218 a(n) = Sum_{k>=1} (3*k - 1)^n / 2^k.

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A355219 a(n) = Sum_{k>=1} (4*k - 2)^n / 2^k.

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A355220 a(n) = Sum_{k>=1} (4*k - 1)^n / 2^k.

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A052825 A simple grammar: partial sums of A008965.

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A052861 E.g.f.: log((1-x)/(1-2*x))*x/(1-x).

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A080163 Sum of an infinite series: a(n) = Sum_{k>=0} ((k+1)*(k+2))^n/(16*(2^k)).

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A052861 E.g.f.: log((1-x)/(1-2x))x/(1-x).

A080163 Sum of an infinite series: a(n) = Sum_{k>=0} ((k+1)(k+2))^n/(16(2^k)).