A134055 -id:A134055 - cp's OEIS frontend

A218672 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(nx)^n/n! exp(-nxA(n*x)).

1, 1, 2, 9, 63, 659, 9833, 206961, 6133990, 256650268, 15213478000, 1281205909177, 153588353066135, 26245044813624300, 6399076697684238375, 2227912079081482302977, 1108302173165578509079527, 788171767077184315422131588, 801638519723021288783092512047

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare to the LambertW identities:
(1) Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).
(2) Sum_{n>=0} n^n * x^n * C(x)^n/n! * exp(-n*x*C(x)) = C(x), where C(x) = 1 + x*C(x)^2 is the o.g.f. of the Catalan numbers (A000108).

			O.g.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 63*x^4 + 659*x^5 + 9833*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2*x)^2/2!*exp(-2*x*A(2*x)) + 3^3*x^3*A(3*x)^3/3!*exp(-3*x*A(3*x)) + 4^4*x^4*A(4*x)^4/4!*exp(-4*x*A(4*x)) + 5^5*x^5*A(5*x)^5/5!*exp(-5*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Paul D. Hanna, Table of n, a(n) for n = 0..50

Cf. A218673, A218674, A218675, A218676, A218681.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.
Cf. A193363, A221409, A221410, A221411, A221412, A221413.
Cf. A209276, A209277.

a[n_] := Module[{A}, A[x_] = 1 + x; For[i = 1, i <= n, i++, A[x_] = Sum[If[k == 0, 1, k^k] x^k A[k x]^k/k! Exp[-k x A[k x] + x O[x]^i] // Normal, {k, 0, n}]]; Coefficient[ A[x], x, n]];
a /@ Range[0, 18] (* Jean-François Alcover, Sep 29 2019 *)

{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A,x,k*x)^k/k!*exp(-k*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A218670 O.g.f.: Sum_{n>=0} n^n * (1+nx)^n x^n/n! * exp(-nx(1+n*x)).

Original entry on oeis.org

1, 1, 2, 7, 26, 116, 556, 2927, 16388, 97666, 612136, 4023878, 27579410, 196537134, 1451102836, 11074811191, 87160086800, 706055915318, 5876662642720, 50182337830986, 439036984440316, 3930618736372336, 35970734643745496, 336153100655220126, 3205000520319374116

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare the o.g.f. to the curious identity:

1/(1-x^2) = Sum_{n>=0} (1+n*x)^n * x^n/n! * exp(-x*(1+n*x)).

			O.g.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 26*x^4 + 116*x^5 + 556*x^6 + 2927*x^7 +...
where
A(x) = 1 + (1+x)*x*exp(-x*(1+x)) + 2^2*(1+2*x)^2*x^2/2!*exp(-2*x*(1+2*x)) + 3^3*(1+3*x)^3*x^3/3!*exp(-3*x*(1+3*x)) + 4^4*(1+4*x)^4*x^4/4!*exp(-4*x*(1+4*x)) + 5^5*(1+5*x)^5*x^5/5!*exp(-5*x*(1+5*x)) +...
simplifies to a power series in x with integer coefficients.

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Cf. A218677, A218678, A218679, A218667, A218668, A218669, A134055, A218671, A217900.

{a(n)=local(A=1+x);A=sum(k=0,n,k^k*(1+k*x)^k*x^k/k!*exp(-k*x*(1+k*x)+x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A174845 O.g.f.: Sum_{n>=0} n^(2n) x^n / (1 - n^2x)^n exp( -n^2x / (1 - n^2x) ) / n!.

Original entry on oeis.org

1, 1, 8, 153, 4981, 236970, 15211158, 1250791640, 127078235560, 15531504729378, 2237017556966100, 373533515381767037, 71351421971134445583, 15419725101750288678775, 3734978285744386546427032, 1005908662614385539285407741, 299140901286981469075716747245

Offset: 0

Paul D. Hanna, Nov 06 2012

nonn

			O.g.f.: A(x) = 1 + x + 8*x^2 + 153*x^3 + 4981*x^4 + 236970*x^5 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^4*x^2/(1-2^2*x)^2*exp(-2^2*x/(1-2^2*x))/2! + 3^6*x^3/(1-3^2*x)^3*exp(-3^2*x/(1-3^2*x))/3! + 4^8*x^4/(1-4^2*x)^4*exp(-4^2*x/(1-4^2*x))/4! +...
simplifies to a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..268

Cf. A134055, A222053, A222054, A243942, A008277.

Flatten[{1,Table[Sum[Binomial[n-1,k-1] * StirlingS2[2*n,k],{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Aug 11 2014 *)
a[ n_] := SeriesCoefficient[ 1 + Sum[(k^2 x)^k / (1 - k^2 x)^k Exp[-k^2 x / (1 - k^2 x)] / k!, {k, n + 1}], {x, 0, n}]; (* Michael Somos, Jun 27 2017 *)

a(n)=polcoeff(sum(k=0, n+1, (k^2*x)^k/(1-k^2*x)^k*exp(-k^2*x/(1-k^2*x+x*O(x^n)))/k!), n) \\ Paul D. Hanna, Nov 04 2012
for(n=0, 25, print1(a(n), ", "))

Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)
{a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1) * Stirling2(2*n, k)))}
for(n=0,25,print1(a(n),", "))\\ Paul D. Hanna, Mar 08 2013

a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(2*n,k) for n>0 with a(0)=1. - Paul D. Hanna, Mar 08 2013

A218673 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^nA(nx)^(2n)/n! exp(-nxA(n*x)^2).

Original entry on oeis.org

1, 1, 3, 20, 209, 3173, 67292, 1970761, 79764057, 4490097388, 354111363537, 39360693851404, 6193012446752244, 1383433132321835172, 439684769985895688173, 199116777197880585373014, 128631139424158036273736167, 118640007280899188486618513612

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare to the LambertW identity:

Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

			O.g.f.: A(x) = 1 + x + 3*x^2 + 20*x^3 + 209*x^4 + 3173*x^5 + 67292*x^6 +...
where
A(x) = 1 + x*A(x)^2*exp(-x*A(x)^2) + 2^2*x^2*A(2*x)^4/2!*exp(-2*x*A(2*x)^2) + 3^3*x^3*A(3*x)^6/3!*exp(-3*x*A(3*x)^2) + 4^4*x^4*A(4*x)^8/4!*exp(-4*x*A(4*x)^2) + 5^5*x^5*A(5*x)^10/5!*exp(-5*x*A(5*x)^2) +...
simplifies to a power series in x with integer coefficients.

Cf. A218672, A218674, A218675, A218676.

Cf. A217900, A218670, A218667, A218668, A218669, A134055.

{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^2,x,k*x)^k/k!*exp(-k*x*subst(A^2,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A218674 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^nA(nx)^(3n)/n! exp(-nxA(n*x)^3).

Original entry on oeis.org

1, 1, 4, 34, 455, 8710, 230077, 8285224, 407456797, 27587687551, 2596034329278, 342275007167359, 63606742005546232, 16730509857101195808, 6246818082857455197662, 3317816101992338134691233, 2510420393373091580780786808, 2709148467943025007607468405672

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare to the LambertW identity:

Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

			O.g.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 455*x^4 + 8710*x^5 + 230077*x^6 +...
where
A(x) = 1 + x*A(x)^3*exp(-x*A(x)^3) + 2^2*x^2*A(2*x)^6/2!*exp(-2*x*A(2*x)^3) + 3^3*x^3*A(3*x)^9/3!*exp(-3*x*A(3*x)^3) + 4^4*x^4*A(4*x)^12/4!*exp(-4*x*A(4*x)^3) + 5^5*x^5*A(5*x)^15/5!*exp(-5*x*A(5*x)^3) +...
simplifies to a power series in x with integer coefficients.

Cf. A218672, A218673, A218675, A218676.

Cf. A217900, A218670, A218667, A218668, A218669, A134055.

{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^3,x,k*x)^k/k!*exp(-k*x*subst(A^3,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A218675 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^nA(nx)^(4n)/n! exp(-nxA(n*x)^4).

Original entry on oeis.org

1, 1, 5, 51, 817, 18562, 576687, 24203258, 1375038677, 106708683355, 11435867474152, 1708844338589752, 358640659116617571, 106261016900832212139, 44607231638918264608274, 26598477338494285370797703, 22569718290467849884279856477

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare to the LambertW identity:

Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

			O.g.f.: A(x) = 1 + x + 5*x^2 + 51*x^3 + 817*x^4 + 18562*x^5 + 576687*x^6 +...
where
A(x) = 1 + x*A(x)^4*exp(-x*A(x)^4) + 2^2*x^2*A(2*x)^8/2!*exp(-2*x*A(2*x)^4) + 3^3*x^3*A(3*x)^12/3!*exp(-3*x*A(3*x)^4) + 4^4*x^4*A(4*x)^16/4!*exp(-4*x*A(4*x)^4) + 5^5*x^5*A(5*x)^20/5!*exp(-5*x*A(5*x)^4) +...
simplifies to a power series in x with integer coefficients.

Cf. A218672, A218673, A218674, A218676.

Cf. A217900, A218670, A218667, A218668, A218669, A134055.

{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^4,x,k*x)^k/k!*exp(-k*x*subst(A^4,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A218676 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^nA(nx)^(5n)/n! exp(-nxA(n*x)^5).

Original entry on oeis.org

1, 1, 6, 71, 1311, 34146, 1207717, 57298282, 3653975784, 316252925221, 37596625187796, 6206102367103899, 1434418185304457039, 466995106832397752352, 215051811411620578152401, 140491107719613466192347681, 130481943378389095603359529403

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare to the LambertW identity:

Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

			O.g.f.: A(x) = 1 + x + 6*x^2 + 71*x^3 + 1311*x^4 + 34146*x^5 + 1207717*x^6 +...
where
A(x) = 1 + x*A(x)^5*exp(-x*A(x)^5) + 2^2*x^2*A(2*x)^10/2!*exp(-2*x*A(2*x)^5) + 3^3*x^3*A(3*x)^15/3!*exp(-3*x*A(3*x)^5) + 4^4*x^4*A(4*x)^20/4!*exp(-4*x*A(4*x)^5) + 5^5*x^5*A(5*x)^25/5!*exp(-5*x*A(5*x)^5) +...
simplifies to a power series in x with integer coefficients.

Cf. A218672, A218673, A218674, A218675.

Cf. A217900, A218670, A218667, A218668, A218669, A134055.

{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^5,x,k*x)^k/k!*exp(-k*x*subst(A^5,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A222053 O.g.f.: Sum_{n>=0} (n^3x)^n/(1-n^3x)^n * exp(-n^3x/(1-n^3x)) / n!.

Original entry on oeis.org

1, 1, 32, 3536, 877221, 394506859, 284110844070, 302350295364613, 449340338669205876, 894210483750815778132, 2306748823711254973903838, 7516588630649080782251419791, 30292392269310179039574629318038, 148358895760995636729844370111255773

Offset: 0

Paul D. Hanna, Mar 08 2013

nonn

			O.g.f.: A(x) = 1 + x + 32*x^2 + 3536*x^3 + 877221*x^4 + 394506859*x^5 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^6*x^2/(1-2^3*x)^2*exp(-2^3*x/(1-2^3*x))/2! + 3^9*x^3/(1-3^3*x)^3*exp(-3^3*x/(1-3^3*x))/3! + 4^12*x^4/(1-4^3*x)^4*exp(-4^3*x/(1-4^3*x))/4! +...
simplifies to a power series in x with integer coefficients.

Cf. A134055, A174845, A222054, A217913, A008277.

{a(n)=polcoeff(sum(k=0, n+1, (k^3*x)^k/(1-k^3*x)^k*exp(-k^3*x/(1-k^3*x+x*O(x^n)))/k!), n)}
for(n=0, 25, print1(a(n), ", "))

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1) * Stirling2(3*n, k)))}
for(n=0, 25, print1(a(n), ", "))

a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(3*n,k) for n>0 with a(0)=1.

A222054 O.g.f.: Sum_{n>=0} (n^4x)^n/(1-n^4x)^n * exp(-n^4x/(1-n^4x)) / n!.

Original entry on oeis.org

1, 1, 128, 90621, 193322261, 933620289929, 8632521193856886, 136885314823146617517, 3443427946573913689696192, 129667338445150206244162849650, 6988095504452769015520539806767120, 520011535068804196524689647521015780176

Offset: 0

Paul D. Hanna, Mar 08 2013

nonn

			O.g.f.: A(x) = 1 + x + 128*x^2 + 90621*x^3 + 193322261*x^4 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^8*x^2/(1-2^4*x)^2*exp(-2^4*x/(1-2^4*x))/2! + 3^12*x^3/(1-3^4*x)^3*exp(-3^4*x/(1-3^4*x))/3! + 4^16*x^4/(1-4^4*x)^4*exp(-4^4*x/(1-4^4*x))/4! +...
simplifies to a power series in x with integer coefficients.

Cf. A134055, A174845, A222053, A217913, A008277.

{a(n)=polcoeff(sum(k=0, n+1, (k^4*x)^k/(1-k^4*x)^k*exp(-k^4*x/(1-k^4*x+x*O(x^n)))/k!), n)}
for(n=0, 20, print1(a(n), ", "))

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1) * Stirling2(4*n, k)))}
for(n=0, 20, print1(a(n), ", "))

a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(4*n,k) for n>0 with a(0)=1.

A243942 O.g.f.: Sum_{n>=0} n^(2n) x^n / (1 - nx)^n exp( -n^2x / (1 - nx) ) / n!.

Original entry on oeis.org

1, 1, 8, 121, 2698, 79654, 2929238, 129004633, 6619919386, 387904397222, 25555935470016, 1869945551975658, 150459006927310348, 13203459856456213172, 1254972882696473807298, 128439184335788533011489, 14082139161229781077548346, 1646731810035799151750487814

Offset: 0

Paul D. Hanna, Aug 09 2014

nonn

			O.g.f.: A(x) = 1 + x + 8*x^2 + 121*x^3 + 2698*x^4 + 79654*x^5 + 2929238*x^6 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^4*x^2/(1-2*x)^2*exp(-4*x/(1-2*x))/2! + 3^6*x^3/(1-3*x)^3*exp(-9*x/(1-3*x))/3! + 4^8*x^4/(1-4*x)^4*exp(-16*x/(1-4*x))/4! + 5^10*x^5/(1-5*x)^5*exp(-25*x/(1-5*x))/5! +...
simplifies to a power series in x with integer coefficients.
Illustrate the terms by:
a(1) = 1*1 = 1;
a(2) = 1*1 + 1*7 = 8;
a(3) = 1*1 + 2*15 + 1*90 = 121;
a(4) = 1*1 + 3*31 + 3*301 + 1*1701 = 2698;
a(5) = 1*1 + 4*63 + 6*966 + 4*7770 + 1*42525 = 79654; ...
where Stirling2(n+k,k) forms a rectangular table as follows:
1, 1,  1,   1,    1,     1,      1,      1, ...;
0, 1,  3,   6,   10,    15,     21,     28, ...;
0, 1,  7,  25,   65,   140,    266,    462, ...;
0, 1, 15,  90,  350,  1050,   2646,   5880, ...;
0, 1, 31, 301, 1701,  6951,  22827,  63987, ...;
0, 1, 63, 966, 7770, 42525, 179487, 627396, ...; ...

Cf. A134055, A174845, A218667, A048993 (Stirling2).

Flatten[{1,Table[Sum[Binomial[n-1,k-1] * StirlingS2[n+k,k],{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Aug 11 2014 *)

{a(n)=polcoeff(sum(k=0, n+1, (k^2*x)^k/(1-k*x)^k*exp(-k^2*x/(1-k*x+x*O(x^n)))/k!), n)}
for(n=0, 25, print1(a(n), ", "))

{Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!}
{a(n)=if(n==0, 1, sum(k=1, n, Stirling2(n+k, k) * binomial(n-1, k-1)))}
for(n=0, 30, print1(a(n), ", "))

a(n) = Sum_{k=1..n} C(n-1,k-1) * Stirling2(n+k,k) for n>0, a(0)=1.

a(n) = c * (r^2/((1-r)*(2*r-1)))^n * n^(n-1/2) / exp(n), where r = 0.859294411517830517100430385442711799997876163... is the root of the equation (1-r)*(1+r)/r^2 = -LambertW(-exp(-1-1/r)*(1+r)/r), and c = 0.4180257159270405799046057130547446708890452... . - Vaclav Kotesovec, Aug 11 2014

cp's OEIS Frontend

A218672 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n*x)^n/n! * exp(-n*x*A(n*x)).

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A218670 O.g.f.: Sum_{n>=0} n^n * (1+n*x)^n * x^n/n! * exp(-n*x*(1+n*x)).

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A174845 O.g.f.: Sum_{n>=0} n^(2*n) * x^n / (1 - n^2*x)^n * exp( -n^2*x / (1 - n^2*x) ) / n!.

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A218673 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(2*n)/n! * exp(-n*x*A(n*x)^2).

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A218674 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(3*n)/n! * exp(-n*x*A(n*x)^3).

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A218675 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(4*n)/n! * exp(-n*x*A(n*x)^4).

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A218676 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(5*n)/n! * exp(-n*x*A(n*x)^5).

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A222053 O.g.f.: Sum_{n>=0} (n^3*x)^n/(1-n^3*x)^n * exp(-n^3*x/(1-n^3*x)) / n!.

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A218672 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(nx)^n/n! exp(-nxA(n*x)).

A218670 O.g.f.: Sum_{n>=0} n^n * (1+nx)^n x^n/n! * exp(-nx(1+n*x)).

A174845 O.g.f.: Sum_{n>=0} n^(2n) x^n / (1 - n^2x)^n exp( -n^2x / (1 - n^2x) ) / n!.

A218673 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^nA(nx)^(2n)/n! exp(-nxA(n*x)^2).

A218674 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^nA(nx)^(3n)/n! exp(-nxA(n*x)^3).

A218675 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^nA(nx)^(4n)/n! exp(-nxA(n*x)^4).

A218676 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^nA(nx)^(5n)/n! exp(-nxA(n*x)^5).

A222053 O.g.f.: Sum_{n>=0} (n^3x)^n/(1-n^3x)^n * exp(-n^3x/(1-n^3x)) / n!.

A222054 O.g.f.: Sum_{n>=0} (n^4x)^n/(1-n^4x)^n * exp(-n^4x/(1-n^4x)) / n!.

A243942 O.g.f.: Sum_{n>=0} n^(2n) x^n / (1 - nx)^n exp( -n^2x / (1 - nx) ) / n!.