A136658 -id:A136658 - cp's OEIS frontend

A049376 Row sums of triangle A046089.

1, 1, 4, 22, 154, 1306, 12976, 147484, 1883932, 26680924, 414468496, 7001104936, 127677078904, 2498712779512, 52209534323584, 1159559538626896, 27269218041047056, 676732851527182864, 17669429275516846912, 484087943980439097184, 13882791112964223876256

Offset: 0

Wolfdieter Lang

nonn

a(n) is the number of n-permutations where each cycle has two (not necessarily distinct) roots. Here a root means a designated element in a cycle. Cf. A000262 which gives the number of n-permutations with a single root in each cycle. Note that the order of designating the elements is not important. Cf. (A bijection from endofunctions to "doubly" rooted trees where the order of designating the roots is important) Miklos Bona, A Walk Through Combinatorics, World Scientific Publishing, 2006, page 216. - Geoffrey Critzer, May 17 2012.

			a(2) = 4 because we have: (1'')(2'');(1''2);(12'');(1'2') where the permutations are given in cycle notation and the two roots in each cycle are designated by a '.

Alois P. Heinz, Table of n, a(n) for n = 0..436
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column k=3 of A291709.
Cf. A000085, A000262, A046089, A136658.
Cf. A004211, A375173.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+1)!/2*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017
a := proc(n) option remember; `if`(n < 3, [1, 1, 4][n + 1],
a(n-1)*(3*n-2) - a(n-2)*3*(n-1)*(n-2) + a(n-3)*(n-1)*(n-2)*(n-3)) end:
seq(a(n), n=0..20); # after Emanuele Munarini, Peter Luschny, Sep 09 2017

nn = 15;Drop[Range[0, nn]! CoefficientList[Series[Exp[x/(1 - x) + x^2/2/(1 - x)^2], {x, 0, nn}], x], 1]  (* Geoffrey Critzer, May 17 2012 *)

E.g.f.: exp(p(x)) with p(x) := x*(2-x)/(2*(1-x)^2) (E.g.f. first column of A046089).
Lah transform of A000085: a(n) = Sum_{k=0..n} n!/k!*binomial(n-1,k-1) * A000085(k). - Vladeta Jovovic, Oct 02 2003
a(n+3) - (3*n+7)*a(n+2) + 3*(n+1)*(n+2)*a(n+1) - n*(n+1)*(n+2)* a(n) = 0. - Emanuele Munarini, Sep 08 2017
a(n) ~ n^(n-1/6) / sqrt(3) * exp(-1/3 + n^(1/3)/2 + 3*n^(2/3)/2 - n). - Vaclav Kotesovec, Oct 23 2017
E.g.f.: Sum_{n>=0} ( Integral 1/(1-x)^3 dx )^n / n!, where the constant of integration is taken to be zero. - Paul D. Hanna, Apr 27 2019
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004211(k).
a(n) = (1/exp(1/2)) * (-1)^n * n! * Sum_{k>=0} binomial(-2*k,n)/(2^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A294046 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(1/(1-x)^k - 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 13, 0, 1, 4, 21, 68, 73, 0, 1, 5, 36, 195, 580, 501, 0, 1, 6, 55, 424, 2241, 5912, 4051, 0, 1, 7, 78, 785, 6136, 30483, 69784, 37633, 0, 1, 8, 105, 1308, 13705, 104544, 476469, 933200, 394353, 0, 1, 9, 136, 2023, 26748

Offset: 0

Seiichi Manyama, Oct 22 2017

			Square array A(n,k) begins:
   1,   1,    1,     1,      1, ...
   0,   1,    2,     3,      4, ...
   0,   3,   10,    21,     36, ...
   0,  13,   68,   195,    424, ...
   0,  73,  580,  2241,   6136, ...
   0, 501, 5912, 30483, 104544, ...

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

Columns k=0..5 give A000007, A000262, A136658, A202826, A294050, A294051.
Rows n=0..2 give A000012, A001477, A014105.
Main diagonal gives A294047.
Cf. A291709.

A[0, ] = 1; A[n, k_] := k*(n-1)!*Sum[Binomial[j+k-1, k]*A[n-j, k]/(n-j)!, {j, 1, n}];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

A(0,k) = 1 and A(n,k) = k * (n-1)! * Sum_{j=1..n} binomial(j+k-1,k)*A(n-j,k)/(n-j)! for n > 0.

A346417 E.g.f.: exp(exp(2*(exp(x) - 1)) - 1).

Original entry on oeis.org

1, 2, 10, 66, 538, 5186, 57402, 714594, 9853978, 148774914, 2436823034, 42979319202, 811254807770, 16302732719682, 347248840767162, 7809649226242530, 184831773033020826, 4589793199157616770, 119272846472231229818, 3235960069037751550498, 91466308730323104617050

Offset: 0

Ilya Gutkovskiy, Jul 16 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..449

Cf. A000258, A000898, A001861, A055882, A126390, A136658.

b:= proc(n, t, m) option remember; `if`(n=0, `if`(t=1, 1,
      b(m, 1, 0)*2^m) , m*b(n-1, t, m)+b(n-1, t, m+1))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 06 2021

nmax = 20; CoefficientList[Series[Exp[Exp[2 (Exp[x] - 1)] - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 2^k BellB[k], {k, 0, n}], {n, 0, 20}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] BellB[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(2*(exp(x) - 1)) - 1))) \\ Michel Marcus, Jul 19 2021

a(n) = Sum_{k=0..n} Stirling2(n,k) * 2^k * Bell(k).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A001861(k) * a(n-k).

A202826 E.g.f.: exp( 1/(1-x)^3 - 1 ).

Original entry on oeis.org

1, 3, 21, 195, 2241, 30483, 476469, 8383203, 163532385, 3496040163, 81159271029, 2030708891907, 54427341596769, 1554460972941555, 47097454520401749, 1507969940021725347, 50850987639474121281, 1800630391669594010307, 66775808799868618561365

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 3*x + 21*x^2/2! + 195*x^3/3! + 2241*x^4/4! +...
where
log(A(x)) = 3*x + 6*x^2 + 10*x^3 + 15*x^4 + 21*x^5 + 28*x^6 +...

Seiichi Manyama, Table of n, a(n) for n = 0..420 (terms 0..200 from Vincenzo Librandi)

Cf. A000262, A136658, A202824.

CoefficientList[Series[E^(1/(1-x)^3-1), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 12 2013 *)
Table[Sum[Abs[StirlingS1[n, k]] 3^k BellB[k], {k, 0, n}], {n, 0, 100}] (* Emanuele Munarini, Sep 01 2017 *)

makelist(sum(abs(stirling1(n,k))*3^k*belln(k),k,0,n),n,0,12); /* Emanuele Munarini, Sep 01 2017 */

{a(n)=n!*polcoeff(exp(1/(1-x +x*O(x^n))^3-1),n)}

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k) *(-1)^(n-k)*3^k)}

a(n) = Sum_{k=0..n} Stirling1(n, k) * Bell(k) * (-1)^(n-k) * 3^k.
a(n) ~ n! * 1/2*3^(1/8)*exp(sqrt(3*n)/2 -3/4 + (3*n)^(1/4)*(4/3*sqrt(n) + 5/24*sqrt(3)) )/(sqrt(2*Pi)*n^(5/8)) * (1 + 871/2560*(3/n)^(1/4)). - Vaclav Kotesovec, Feb 12 2013
a(n+4) - (4*n+15)*a(n+3) + 6*(n+2)*(n+3)*a(n+2) - 4*(n+1)*(n+2)+(n+3)*a(n+1) + n*(n+1)*(n+2)*(n+3)*a(n) = 0. - Emanuele Munarini, Sep 01 2017

Example corrected by Vaclav Kotesovec, Feb 12 2013

A294189 E.g.f.: exp(2*(1/(1-x)^2 - 1)).

Original entry on oeis.org

1, 4, 28, 256, 2848, 37024, 547936, 9064192, 165339904, 3290839552, 70870959616, 1640130678784, 40555349598208, 1066271901822976, 29684252411219968, 871864036043259904, 26931181039027879936, 872418396180001718272, 29564373329109844885504

Offset: 0

Seiichi Manyama, Oct 24 2017

Seiichi Manyama, Table of n, a(n) for n = 0..427

Column k=2 of A294188.

Cf. A136658.

nmax = 20; CoefficientList[Series[E^(2*(1/(1-x)^2 - 1)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 28 2025 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(2*(1/(1-x)^2-1))))

From Vaclav Kotesovec, Aug 28 2025: (Start)
a(n) = (3*n+1)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-3).
a(n) ~ 2^(1/3) * 3^(-1/2) * exp(-4/3 + 2^(1/3)*n^(1/3) + 3*2^(-1/3)*n^(2/3) - n) * n^(n - 1/6) * (1 + 19/(27*2^(1/3)*n^(1/3)) - 11/(3645*2^(2/3)*n^(2/3))). (End)

A346432 a(0) = 1; a(n) = n! * Sum_{k=0..n-1} (n-k+1) * a(k) / k!.

Original entry on oeis.org

1, 2, 14, 144, 1968, 33600, 688320, 16450560, 449326080, 13806858240, 471395635200, 17703899136000, 725338710835200, 32193996432998400, 1538840509503897600, 78808952068374528000, 4305129487814098944000, 249876735246162984960000, 15356385691181506363392000

Offset: 0

Ilya Gutkovskiy, Jul 17 2021

nonn

Cf. A000670, A001339, A002866, A003480, A007840, A052555, A052567, A136658, A216794, A308939, A346433.

a[0] = 1; a[n_] := a[n] = n! Sum[(n - k + 1) a[k]/k!, {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 - 1/(1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) StirlingS1[n, k] 2^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 0, 18}]

my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - 1 / (1 - x)^2))) \\ Michel Marcus, Jul 18 2021

E.g.f.: 1 / (2 - 1 / (1 - x)^2).
E.g.f.: 1 / (1 - Sum_{k>=1} (k+1) * x^k).
a(0) = 1, a(1) = 2, a(2) = 14; a(n) = 4 * n * a(n-1) - 2 * n * (n-1) * a(n-2).
a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n,k) * 2^k * A000670(k).
a(n) = n! * A003480(n).

A355267 Triangle read by rows, T(n, k) = n! * [y^k] [x^n] exp(1/(1 - x)^(1 + y) - 1), for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 3, 5, 2, 13, 29, 21, 5, 73, 200, 202, 90, 15, 501, 1609, 2045, 1295, 410, 52, 4051, 14809, 22418, 18085, 8220, 1998, 203, 37633, 153453, 267400, 259175, 151165, 53095, 10402, 877, 394353, 1767240, 3463612, 3889620, 2740885, 1241632, 353178, 57676, 4140

Offset: 0

Peter Luschny, Jul 05 2022

			Triangle T(n, k) begins:
[0]      1;
[1]      1,      1;
[2]      3,      5,      2;
[3]     13,     29,     21,      5;
[4]     73,    200,    202,     90,     15;
[5]    501,   1609,   2045,   1295,    410,    52;
[6]   4051,  14809,  22418,  18085,   8220,  1998,   203;
[7]  37633, 153453, 267400, 259175, 151165, 53095, 10402, 877;

Cf. A136658 (row sums), A000007 (alternating row sums), A000262 (column 0), A216313 (column 1), A000110 (main diagonal).

Cf. A355260.

egf := exp(1/(1 - x)^(1 + y) - 1):
ser := series(egf, x, 12): cfx := n -> coeff(ser, x, n):
seq(print(seq(n!*(coeff(cfx(n), y, k)), k = 0..n)), n = 0..8);

cp's OEIS Frontend

A049376 Row sums of triangle A046089.

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A294046 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(1/(1-x)^k - 1).

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A346417 E.g.f.: exp(exp(2*(exp(x) - 1)) - 1).

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A202826 E.g.f.: exp( 1/(1-x)^3 - 1 ).

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A294189 E.g.f.: exp(2*(1/(1-x)^2 - 1)).

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A346432 a(0) = 1; a(n) = n! * Sum_{k=0..n-1} (n-k+1) * a(k) / k!.

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A355267 Triangle read by rows, T(n, k) = n! * [y^k] [x^n] exp(1/(1 - x)^(1 + y) - 1), for 0 <= k <= n.

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