A375173 -id:A375173 - cp's OEIS frontend

A080893 Expansion of e.g.f. exp(xC(x)) = exp((1-sqrt(1-4x))/2), where C(x) is the g.f. of the Catalan numbers A000108.

1, 1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353, 28875761731, 1098127402131, 46150226651233, 2124008553358849, 106246577894593683, 5739439214861417731, 332993721039856822081, 20651350143685984386753

Offset: 0

Emanuele Munarini, Mar 31 2003

Essentially the same as A001517: a(n+1) = A001517(n).

For k >= 2, the difference a(n+k) - a(n) is divisible by k. It follows that for each k, the sequence formed by taking a(n) modulo k is periodic with period dividing k. For example, modulo 10 the sequence becomes [1, 1, 3, 9, 3, 1, 1, 3, 9, 3, ...], a purely periodic sequence of period 5. Cf. A047974. - Peter Bala, Feb 11 2025

W. Mlotkowski and A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A000108, A001517, A009235, A375173.

y[x_] := y[x] = 2(2x - 3)y[x - 1] + y[x - 2]; y[0] = 1; y[1] = 1; Table[y[n],{n,0,17}]
With[{nn=20},CoefficientList[Series[Exp[(1-Sqrt[1-4x])/2],{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Oct 30 2011 *)

{a(n) = if( n<1, n = 1 - n); n! * polcoeff( exp( (1 - sqrt(1 - 4*x + x * O(x^n))) / 2), n)} /* Michael Somos, Apr 07 2012 */

A080893 = lambda n: hypergeometric([-n+1, n], [], -1)
[simplify(A080893(n)) for n in (0..19)] # Peter Luschny, Oct 17 2014

E.g.f.: exp((1-sqrt(1-4*x))/2).
D-finite with recurrence: a(n+2) = 2*(2*n + 1)*a(n+1) + a(n).
Recurrence: y(n+1) = Sum_{k = 0..n} binomial(n, k)*binomial(2k, k)*k!*y(n-k).
a(1 - n) = a(n).  a(n + 1) = A001517(n). - Michael Somos, Apr 07 2012
G.f.: 1 + x/Q(0), where Q(k)= 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
a(n) ~ 2^(2*n-3/2)*n^(n-1)/exp(n-1/2). - Vaclav Kotesovec, Jun 26 2013
a(n) = hypergeom([-n+1, n], [], -1). - Peter Luschny, Oct 17 2014
a(n) = Sum_{k=0..n} (-4)^(n-k) * Stirling1(n,k) * A009235(k) = (-4)^n * Sum_{k=0..n} (1/2)^k * Stirling1(n,k) * Bell_k(-1/2), where Bell_n(x) is n-th Bell polynomial. - Seiichi Manyama, Aug 02 2024

A049376 Row sums of triangle A046089.

Original entry on oeis.org

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

A380257 Expansion of e.g.f. exp( (1/(1-3*x)^(2/3) - 1)/2 ).

Original entry on oeis.org

1, 1, 6, 56, 706, 11186, 213156, 4742256, 120571676, 3447128796, 109427729096, 3818008773536, 145196289453656, 5976489668054296, 264685744187399536, 12548508890339297856, 634022724191046592016, 34007862777419093053456, 1929842567333195106456416

Offset: 0

Seiichi Manyama, Jan 18 2025

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A049376, A375173, A380258.

Cf. A004211, A380214.

CoefficientList[Series[Exp[ (1/(1-3*x)^(2/3) - 1)/2 ],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Mar 31 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-3*x)^(2/3)-1)/2)))

a(n) = Sum_{k=0..n} 3^(n-k) * |Stirling1(n,k)| * A004211(k) = Sum_{k=0..n} 2^k * 3^(n-k) * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

a(n) = (1/exp(1/2)) * (-3)^n * n! * Sum_{k>=0} binomial(-2*k/3,n)/(2^k * k!).

A375175 Expansion of e.g.f. exp( (exp( (exp(4*x) - 1)/2 ) - 1)/2 ).

Original entry on oeis.org

1, 1, 7, 63, 713, 9753, 156111, 2858103, 58845105, 1344371793, 33713484151, 919838859151, 27105053988793, 857310780134825, 28953291147179007, 1039373409620929671, 39505610599553955809, 1584411299793530257697, 66846625774893448843879

Offset: 0

Seiichi Manyama, Aug 02 2024

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A080893, A375173.

Cf. A004211.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((exp((exp(4*x)-1)/2)-1)/2)))

a(n) = Sum_{k=0..n} 4^(n-k) * Stirling2(n,k) * A004211(k) = 4^n * Sum_{k=0..n} (1/2)^k * Stirling2(n,k) * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

A380258 Expansion of e.g.f. exp( (1/(1-5*x)^(2/5) - 1)/2 ).

Original entry on oeis.org

1, 1, 8, 106, 1954, 46082, 1323064, 44750644, 1741897340, 76672512316, 3764746706176, 203976645319448, 12086590557877144, 777464693554778776, 53948773488864143072, 4016672567726156437744, 319379204127841984947472, 27010128651142535536409360, 2420802590890201251989984128

Offset: 0

Seiichi Manyama, Jan 18 2025

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A049376, A375173, A380257.

Cf. A004211, A025168.

CoefficientList[Series[Exp[ (1/(1-5*x)^(2/5) - 1)/2 ],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Mar 31 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-5*x)^(2/5)-1)/2)))

a(n) = Sum_{k=0..n} 5^(n-k) * |Stirling1(n,k)| * A004211(k) = Sum_{k=0..n} 2^k * 5^(n-k) * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

a(n) = (1/exp(1/2)) * (-5)^n * n! * Sum_{k>=0} binomial(-2*k/5,n)/(2^k * k!).

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A080893 Expansion of e.g.f. exp(x*C(x)) = exp((1-sqrt(1-4*x))/2), where C(x) is the g.f. of the Catalan numbers A000108.

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A049376 Row sums of triangle A046089.

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A380257 Expansion of e.g.f. exp( (1/(1-3*x)^(2/3) - 1)/2 ).

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A375175 Expansion of e.g.f. exp( (exp( (exp(4*x) - 1)/2 ) - 1)/2 ).

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A380258 Expansion of e.g.f. exp( (1/(1-5*x)^(2/5) - 1)/2 ).

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A080893 Expansion of e.g.f. exp(xC(x)) = exp((1-sqrt(1-4x))/2), where C(x) is the g.f. of the Catalan numbers A000108.