A027710 -id:A027710 - cp's OEIS frontend

A078940 Row sums of A078938.

1, 4, 19, 103, 622, 4117, 29521, 227290, 1865881, 16239523, 149142952, 1439618143, 14555631781, 153700654036, 1690684883191, 19328770917499, 229203640111870, 2814018686591089, 35711716110387589, 467766675528462562

Offset: 0

Paul D. Hanna, Dec 18 2002

nonn

Divide by 3^n and insert an initial 1 to get sequence that shifts left one place under 1/3 order binomial transformation. - Franklin T. Adams-Watters, Jul 13 2006

Binomial transform of A027710. - Vaclav Kotesovec, Jun 26 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Column k=3 of A335975.

Cf. A027710, A035009, A078938, A078945, A355254.

A078940 := proc(n) local a,b,i;
a := [seq(2,i=1..n)]; b := [seq(1,i=1..n)];
exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=3,%),66)) end:
seq(A078940(n),n=0..19); # Peter Luschny, Mar 30 2011

Table[n!, {n, 0, 20}]CoefficientList[Series[E^(3E^x-3+x), {x, 0, 20}], x]
Table[1/E^3/3*Sum[m^n/m!*3^m,{m,0,Infinity}],{n,1,20}] (* Vaclav Kotesovec, Mar 12 2014 *)
Table[BellB[n+1, 3]/3, {n, 0, 20}] (* Vaclav Kotesovec, Jan 15 2016 *)
nmax = 20; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - (k+4)*x - 3*(k+1)*x^2/g[k+1]; CoefficientList[Series[1/g[0], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 15 2016, after Sergei N. Gladkovskii *)

E.g.f.: exp(3*(exp(x)-1)+x).
Stirling transform of [1, 3, 3^2, 3^3, ...]. - Gerald McGarvey, Jun 01 2005
Define f_1(x), f_2(x), ... such that f_1(x)=e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n)=e^{-3}*f_n(3). - Milan Janjic, May 30 2008
G.f.: 1/T(0), where T(k) = 1 - (k+4)*x - 3*(k+1)*x^2/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2016
a(n) = exp(-3) * Sum_{k>=0} (k + 1)^n * 3^k / k!. - Ilya Gutkovskiy, Apr 20 2020
a(n) ~ n^(n+1) * exp(n/LambertW(n/3) - n - 3) / (3 * sqrt(1 + LambertW(n/3)) * LambertW(n/3)^(n+1)). - Vaclav Kotesovec, Jun 26 2022
a(0) = 1; a(n) = a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Dec 05 2023

More terms from Robert G. Wilson v, Dec 19 2002

A292860 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 5, 0, 1, 4, 12, 22, 15, 0, 1, 5, 20, 57, 94, 52, 0, 1, 6, 30, 116, 309, 454, 203, 0, 1, 7, 42, 205, 756, 1866, 2430, 877, 0, 1, 8, 56, 330, 1555, 5428, 12351, 14214, 4140, 0, 1, 9, 72, 497, 2850, 12880, 42356, 88563, 89918, 21147, 0

Offset: 0

Seiichi Manyama, Sep 25 2017

			Square array begins:
   1,   1,    1,     1,     1,      1,      1, ...
   0,   1,    2,     3,     4,      5,      6, ...
   0,   2,    6,    12,    20,     30,     42, ...
   0,   5,   22,    57,   116,    205,    330, ...
   0,  15,   94,   309,   756,   1555,   2850, ...
   0,  52,  454,  1866,  5428,  12880,  26682, ...
   0, 203, 2430, 12351, 42356, 115155, 268098, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-10 give: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263, A221159, A276506, A276507.
Rows n=0..2 give A000012, A001477, A002378.
Main diagonal gives A242817.
Same array, different indexing is A189233.
Cf. A292861.

A:= proc(n, k) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017

A[0, ] = 1; A[n /; n >= 0, k_ /; k >= 0] := A[n, k] = k*Sum[Binomial[n-1, j]*A[j, k], {j, 0, n-1}]; A[, ] = 0;
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)
A292860[n_, k_] := BellB[n, k]; Table[A292860[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 23 2021 *)

A(0,k) = 1 and A(n,k) = k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.
A(n,k) = Sum_{j=0..n} k^j * Stirling2(n,j). - Seiichi Manyama, Jul 27 2019
A(n,k) = BellPolynomial(n, k). - Peter Luschny, Dec 23 2021

A221176 a(n) = Sum_{i=0..n} Stirling2(n,i)*2^(4i).

Original entry on oeis.org

1, 16, 272, 4880, 91920, 1810192, 37142288, 791744272, 17490370320, 399558315792, 9421351690000, 228916588400400, 5723078052339472, 147025755978698512, 3876566243300318992, 104789417805394595600, 2901159958960121863952, 82188946843192555474704, 2380551266738846355103504, 70441182699006212824911632

Offset: 0

N. J. A. Sloane, Jan 04 2013

nonn

The number of ways of putting n labeled balls into a set of bags and then putting the bags into 16 labeled boxes. - Peter Bala, Mar 23 2013

Frank Simon, Algebraic Methods for Computing the Reliability of Networks, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. See Table 5.1.

Cf. A000110, A001861, A078944, A221159. A027710, A144180, A144223, A144263, A189233.

With[{nn=20},CoefficientList[Series[Exp[16 (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 19 2024 *)

E.g.f. exp(16*(exp(x) - 1)). - Peter Bala, Mar 23 2013

A357572 Expansion of e.g.f. sinh(sqrt(3) * (exp(x)-1)) / sqrt(3).

Original entry on oeis.org

0, 1, 1, 4, 19, 85, 406, 2191, 13105, 84190, 573121, 4127521, 31434184, 252388957, 2126998693, 18740283556, 172134162631, 1644920020417, 16324076578870, 167938152551491, 1787952325142341, 19667748794844550, 223217829954224029, 2610546296216999197

Offset: 0

Seiichi Manyama, Oct 05 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..562
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A024429, A264037, A357598.

Cf. A027710, A357615, A357737.

a(n) = sum(k=0, (n-1)\2, 3^k*stirling(n, 2*k+1, 2));

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(3))-Bell_poly(n, -sqrt(3)))/(2*sqrt(3)));

a(n) = Sum_{k=0..floor((n-1)/2)} 3^k * Stirling2(n,2*k+1).
a(n) = ( Bell_n(sqrt(3)) - Bell_n(-sqrt(3)) )/(2 * sqrt(3)), where Bell_n(x) is n-th Bell polynomial.
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357615(k).

A343975 a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(k-1).

Original entry on oeis.org

1, 3, 15, 81, 489, 3237, 23211, 178707, 1467051, 12768345, 117263829, 1131901521, 11444383251, 120847326879, 1329303053391, 15197269729689, 180211641841353, 2212525627591533, 28078380387448515, 367782119667874083, 4965441830591976339, 69014083524412401873, 986364827548578356421

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Cf. A027710, A032033, A040027, A078940, A343523, A344735, A344840, A345077, A345078, A345081.

a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 22}]
nmax = 22; A[] = 0; Do[A[x] = 1 + 3 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + 3 * x * A(x/(1 - x)) / (1 - x)^2.

A355254 Expansion of e.g.f. exp(3*(exp(x) - 1) - x).

Original entry on oeis.org

1, 2, 7, 29, 142, 785, 4813, 32240, 233449, 1812161, 14980768, 131174939, 1211111629, 11745451658, 119255234371, 1264050651953, 13952113296766, 160006824960725, 1902825936046105, 23423342243273696, 297982102750214605, 3911917977005948453, 52926119656555824520

Offset: 0

Vaclav Kotesovec, Jun 26 2022

nonn

Inverse binomial transform of A027710.
In general, if m >= 1 and e.g.f. = exp(m*exp(x) + r*x + s) then
a(n) ~ n^(n+r) * exp(n/LambertW(n/m) - n + s) / (m^r * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n+r)).
Equivalently, a(n) ~ n! * (n/m)^r * exp(n/LambertW(n/m) + s) / (sqrt(2*Pi*n * (1 + LambertW(n/m))) * LambertW(n/m)^(n+r)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..545

Cf. A000296, A027710, A078940, A217924.

nmax = 25; CoefficientList[Series[Exp[3*Exp[x]-3-x], {x, 0, nmax}], x] * Range[0, nmax]!

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) - x))) \\ Michel Marcus, Dec 04 2023

a(n) ~ 3 * n^(n-1) * exp(n/LambertW(n/3) - n - 3) / (sqrt(1 + LambertW(n/3)) * LambertW(n/3)^(n-1)).

a(0) = 1; a(n) = -a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Dec 04 2023

A299824 a(n) = (1/e^n)Sum_{j >= 1} j^n n^j / (j-1)!.

Original entry on oeis.org

2, 22, 309, 5428, 115155, 2869242, 82187658, 2661876168, 96202473183, 3838516103310, 167606767714397, 7949901069639228, 407048805012563038, 22376916254447538882, 1314573505901491675965, 82188946843192555474704, 5448870914168179374456623, 381819805747937892412056342

Offset: 1

Pedro Caceres, Feb 19 2018

nonn

For m>1, A242817(m) and a(m-1) are also the m-th and (m+1)-st terms of the sequences "Number of ways of placing X labeled balls into X unlabeled (but (m-1)-colored) boxes". For instance, sequence A144180 for 5-colored boxes (m = 6), has A144180(6) = 12880, and A144180(7) = 115155, which are A242817(6) and a(5) respectively. Same pattern can be observed for A027710, A144223, A144263 (comment added after Omar E. Pol's formula).

			a(4) = (1/e^4)*Sum_{j >= 1} j^4 * 4^j / (j-1)! = 5428.

Alois P. Heinz, Table of n, a(n) for n = 1..368

Cf. A027710, A144180, A144223, A144263, A189233, A242817, A292860.

a(n) = round(exp(-n)*suminf(j = 1, (j^n)*(n^j)/(j-1)!)); \\ Michel Marcus, Feb 24 2018

A299824(n,f=exp(n),S=n/f,t)=for(j=2,oo,S+=(t=j^n*n^j)/(f*=j-1);tn&&return(ceil(S))) \\ For n > 23, use \p## with some ## >= 2n. - M. F. Hasler, Mar 09 2018

a(n) = A189233(n+1,n). - Omar E. Pol, Feb 24 2018
a(n) ~ exp(n/LambertW(1) - 2*n) * n^(n + 1) / (sqrt(1 + LambertW(1)) * LambertW(1)^(n + 1)). - Vaclav Kotesovec, Mar 08 2018
Or: a(n) ~ (1/sqrt(1+w)) * exp(1/w-2)^n * (n/w)^(n+1), with w = LambertW(1) ~ 0.56714329... The relative error decreases from 10^-2 for a(2) to 10^-3 for a(15), but reaches 10^-3.5 only at a(45). - M. F. Hasler, Mar 09 2018

A309084 a(n) = exp(3) * Sum_{k>=0} (-3)^k*k^n/k!.

Original entry on oeis.org

1, -3, 6, -3, -21, 24, 195, -111, -3072, -4053, 57003, 277854, -697539, -12261567, -29861778, 371727465, 3511027599, 2028432480, -188521156857, -1470389129931, 1655487186864, 121873222577823, 915525253963023, -2095901567014530, -103715912230195863, -836215492271268459

Offset: 0

Ilya Gutkovskiy, Jul 11 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..571

Column k = 3 of A292861.

Cf. A000587, A027710, A213170, A309085, A317996.

[1] cat [(&+[((-3)^k*StirlingSecond(m, k)):k in [0..m]]):m in [1..25]]; // Marius A. Burtea, Jul 27 2019

b:= proc(n, m) option remember; `if`(n=0,
      (-3)^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 17 2022

Table[Exp[3] Sum[(-3)^k k^n/k!, {k, 0, Infinity}], {n, 0, 25}]
Table[BellB[n, -3], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Sum[(-3)^j x^j/Product[(1 - k x), {k, 1, j}] , {j, 0, nmax}], {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[Exp[3 (1 - Exp[x])], {x, 0, nmax}], x] Range[0, nmax]!

G.f.: Sum_{j>=0} (-3)^j*x^j / Product_{k=1..j} (1 - k*x).
E.g.f.: exp(3*(1 - exp(x))).
a(n) = Sum_{k=0..n} (-3)^k * Stirling2(n,k).

A367888 Expansion of e.g.f. exp(3(exp(x) - 1) - 2x).

Original entry on oeis.org

1, 1, 4, 13, 61, 304, 1747, 10945, 74830, 550687, 4335109, 36272086, 320980645, 2991373597, 29253607780, 299258487553, 3193634980753, 35469069928792, 409082335024591, 4890313138089133, 60489400453642822, 772967507343358171, 10189818916331129017, 138398721137005215526

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000296, A027710, A126617, A194689, A355254, A367889.

b:= proc(n, k, m) option remember; `if`(n=0, 3^m, `if`(k>0,
      b(n-1, k-1, m+1)*k, 0)+m*b(n-1, k, m)+b(n-1, k+1, m))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 29 2025

nmax = 23; CoefficientList[Series[Exp[3 (Exp[x] - 1) - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + 3 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
Table[Sum[Binomial[n, k] (-2)^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 23}]

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) - 2*x))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 - x * ( 2 * A(x) - 3 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k-2)^n / k!.
a(0) = 1; a(n) = -2 * a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * A027710(k).

A068199 One of a family of sequences that interpolates between the Bell numbers and the factorials.

Original entry on oeis.org

1, 2, 6, 24, 114, 618, 3732, 24702, 177126, 1363740, 11195286, 97437138, 894857712, 8637708858, 87333790686, 922203924216, 10144109299146, 115972625504994, 1375221840671220, 16884112119546534, 214270296662325534, 2806600053170775372, 37892025089041181982

Offset: 0

N. J. A. Sloane, Mar 23 2002

nonn

G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.

Cf. A000110, A001861, this, A068200, A068201, ..., A000142.

Equals 2 * A027710(n).

g:= proc(n) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, k-1)*g(n-k), k=1..n-1))*3)
    end:
a:= n-> `if`(n=0, 1, 2*g(n-1)):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 09 2008

a[n_] := 2*BellB[n-1, 3]; a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 28 2014 *)

E.g.f.: 1 + 2*exp(3exp(x)-3).

More terms from Alois P. Heinz, Oct 09 2008

cp's OEIS Frontend

A078940 Row sums of A078938.

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A292860 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1)).

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A221176 a(n) = Sum_{i=0..n} Stirling2(n,i)*2^(4i).

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A357572 Expansion of e.g.f. sinh(sqrt(3) * (exp(x)-1)) / sqrt(3).

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A343975 a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(k-1).

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A355254 Expansion of e.g.f. exp(3*(exp(x) - 1) - x).

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A299824 a(n) = (1/e^n)*Sum_{j >= 1} j^n * n^j / (j-1)!.

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A309084 a(n) = exp(3) * Sum_{k>=0} (-3)^k*k^n/k!.

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A299824 a(n) = (1/e^n)Sum_{j >= 1} j^n n^j / (j-1)!.

A367888 Expansion of e.g.f. exp(3(exp(x) - 1) - 2x).