cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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)).

Original entry on oeis.org

1, 1, 2, 9, 63, 659, 9833, 206961, 6133990, 256650268, 15213478000, 1281205909177, 153588353066135, 26245044813624300, 6399076697684238375, 2227912079081482302977, 1108302173165578509079527, 788171767077184315422131588, 801638519723021288783092512047
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    {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+n*x)^n * x^n/n! * exp(-n*x*(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

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

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)).

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    {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),", "))

A218667 O.g.f.: Sum_{n>=0} 1/(1-n*x)^n * x^n/n! * exp(-x/(1-n*x)).

Original entry on oeis.org

1, 0, 1, 1, 4, 13, 46, 181, 778, 3585, 17566, 91171, 499324, 2873839, 17317743, 108933098, 713481122, 4855161425, 34257461754, 250177938679, 1887886966690, 14699340919293, 117933068390123, 973776266303732, 8265721830953558, 72052688932613079, 644393453082317301
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare 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)).

Examples

			O.g.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 13*x^5 + 46*x^6 + 181*x^7 +...
where
A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-2*x)^2/2!*exp(-x/(1-2*x)) + x^3/(1-3*x)^3/3!*exp(-x/(1-3*x)) + x^4/(1-4*x)^4/4!*exp(-x/(1-4*x)) + x^5/(1-5*x)^5/5!*exp(-x/(1-5*x)) + x^6/(1-6*x)^6/6!*exp(-x/(1-6*x)) +...
simplifies to a power series in x with integer coefficients.

Links

Crossrefs

Cf. A245111, A245110, A218668, A218669, A218670, A185040, A217900.

Programs

  • PARI
    {a(n)=local(A=1+x,X=x+x*O(x^n));A=sum(k=0,n,1/(1-k*X)^k*x^k/k!*exp(-X/(1-k*X)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* From a(n) = Sum_{k=1..n} Stirling2(n-k, k) * C(-1, k-1) */
{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), ", "))

Formula

a(n) = Sum_{k=1..n} Stirling2(n-k, k) * C(n-1, k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jul 30 2014
Antidiagonal sums of Triangle A245111.

A218669 O.g.f.: Sum_{n>=0} 1/(1-n^3*x)^n * x^n/n! * exp(-x/(1-n^3*x)).

Original entry on oeis.org

1, 0, 1, 7, 97, 1561, 41136, 1551814, 72440460, 4281320257, 324623105584, 30086950057627, 3299720918091511, 428431079916572044, 65637957066642609845, 11659659637028895337265, 2367270866164121777222596, 546795407830461739380895161, 143176487805296033192642234802
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare 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)).

Examples

			O.g.f.: A(x) = 1 + x^2 + 7*x^3 + 97*x^4 + 1561*x^5 + 41136*x^6 +...
where
A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-8*x)^2/2!*exp(-x/(1-8*x)) + x^3/(1-27*x)^3/3!*exp(-x/(1-27*x)) + x^4/(1-64*x)^4/4!*exp(-x/(1-64*x)) + x^5/(1-125*x)^5/5!*exp(-x/(1-125*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218667, A218668, A218670, A217900.

Programs

  • PARI
    {a(n)=local(A=1+x,X=x+x*O(x^n));A=sum(k=0,n,1/(1-k^3*X)^k*x^k/k!*exp(-X/(1-k^3*X)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

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).

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

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

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)).

Examples

			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.

Crossrefs

Cf. A218672, A218674, A218675, A218676.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

  • PARI
    {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^n*A(n*x)^(3*n)/n! * exp(-n*x*A(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

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

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)).

Examples

			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.

Crossrefs

Cf. A218672, A218673, A218675, A218676.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

  • PARI
    {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^n*A(n*x)^(4*n)/n! * exp(-n*x*A(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

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

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)).

Examples

			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.

Crossrefs

Cf. A218672, A218673, A218674, A218676.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

  • PARI
    {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^n*A(n*x)^(5*n)/n! * exp(-n*x*A(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

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

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)).

Examples

			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.

Crossrefs

Cf. A218672, A218673, A218674, A218675.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

  • PARI
    {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),", "))
Showing 1-8 of 8 results.

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